# Multivariate Symmetry: Distribution-Free Testing via Optimal Transport

Zhen Huang
Department of Statistics  Columbia University
e-mail<EMAIL_ADDRESS>
and
Bodhisattva Sen
Department of Statistics Supported by NSF grant DMS-2015376.  Columbia
University
e-mail<EMAIL_ADDRESS>
###### Abstract

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon,
1945) are among the first examples of a nonparametric test. These procedures —
based on signs, (absolute) ranks and signed-ranks — yield distribution-free
tests for symmetry in one-dimension. In this paper we propose a novel and
unified framework for distribution-free testing of multivariate symmetry (that
includes central symmetry, sign symmetry, spherical symmetry, etc.) based on
the theory of optimal transport. Our approach leads to notions of
distribution-free generalized multivariate signs, ranks and signed-ranks. As a
consequence, we develop analogues of the sign and Wilcoxon signed-rank tests
that share many of the appealing properties of their one-dimensional
counterparts. In particular, the proposed tests are exactly distribution-free
in finite samples with an asymptotic normal limit, and adapt to various
notions of multivariate symmetry. We study the consistency of the proposed
tests and their behavior under local alternatives, and show that the proposed
generalized Wilcoxon signed-rank (GWSR) test is particularly powerful against
location shift alternatives. We show that in a large class of such models, our
GWSR test suffers from no loss in (asymptotic) efficiency, when compared to
Hotelling’s $T^{2}$ test, despite being nonparametric and exactly
distribution-free. An appropriately score transformed version of the GWSR
statistic leads to a locally asymptotically optimal test. Further, our method
can be readily used to construct distribution-free confidence sets for the
center of symmetry.

##  1 Introduction

Symmetry is everywhere, from the bilateral symmetry in Greek sculpture, to the
description of crystalline structure in nature by modern group theory [103,
93]. The study of symmetry in statistics also has an old and rich history.
Perhaps the first published report of a nonparametric test goes back to the
landmark paper [4] by John Arbuthnott where he essentially performed a
statistical hypothesis test for univariate symmetry, computing a $p$-value via
the sign test; see e.g., [7, 41]. Some statisticians also date the beginning
of (classical) nonparametric statistics, in particular rank-based methods, to
the appearance of Wilcoxon’s 1945 paper [104], which introduced the Wilcoxon
signed-rank test (WSR) for testing univariate symmetry. Both these classical
tests are distribution-free and have had an enormous impact in the development
of nonparametric statistics.

A univariate random variable $X$ is symmetric (around 0) if $X$ has the same
distribution as $-X$. Given $n$ observations $X_{1},\ldots,X_{n}$ from the
distribution of $X$, the sign test rejects the null hypothesis ${\rm
H}_{0}:X\overset{d}{=}-X$ when $\left|\sum_{i=1}^{n}{\rm sign}(X_{i})\right|$
is large; here ${\rm sign}(X_{i})=1$ if $X_{i}\geq 0$ and is $-1$ otherwise.
In contrast, the Wilcoxon signed-rank test first sorts
$|X_{1}|,\ldots,|X_{n}|$ to obtain their (absolute) ranks $\\{{\rm
rank}(|X_{i}|)\\}_{i=1}^{n}$, and rejects ${\rm H}_{0}$ for large values of
$\left|\sum_{i=1}^{n}{\rm sign}(X_{i}){\rm rank}(|X_{i}|)\right|$. When $X$
has a continuous distribution, the classical sign and signed-rank tests enjoy
several properties that make them powerful and useful in practice, including:
(i) distribution-freeness, (ii) independence of signs and absolute ranks
(under ${\rm H}_{0}$), (iii) simple asymptotic normal limiting distributions,
and (iv) high (relative) efficiency compared to the $t$-test [57, 22]; see
[99, Chapter 15] for a detailed discussion of these results.

It is natural to ask if generalizations of these tests, that satisfy the above
mentioned desirable properties, exist in multi-dimensions. In this paper, we
answer this question in the affirmative by proposing a novel framework for
testing multivariate symmetry, using the theory of optimal transport [100,
101].

While symmetry in one dimension is unambiguous (i.e., $X\overset{d}{=}-X$), in
the multivariate setting there are various notions of symmetry. The most
prominent ones include: (a) central symmetry [91, 13, 80, 83, 32], (b) sign
symmetry [31, 84, 83], and (c) spherical symmetry [6, 39, 52, 2]; a thorough
review of the topic can be found in Serfling, [93].

In this paper, we consider the following general setting for testing
multivariate symmetry. Let $\mathcal{G}$ be a compact subgroup of the
orthogonal group ${\rm O}(p)$ (for $p\geq 1$) — the group of all $p\times p$
real orthogonal matrices, where the group operation is the usual matrix
multiplication. For a random vector ${\bf X}\sim\mu\in\mathcal{P}_{\rm
ac}(\mathbb{R}^{p})$, where $\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$ denotes the
class of all Borel probability measures on $\mathbb{R}^{p}$ with a Lebesgue
density, we are interested in testing the hypothesis of
$\mathcal{G}$-symmetry:

$\textrm{H}_{0}:{\bf X}\stackrel{{\scriptstyle d}}{{=}}Q{\bf X}{\rm\ \ for\
all\ \
}Q\in\mathcal{G}\qquad\qquad\textrm{versus}\qquad\qquad\textrm{H}_{1}:\textrm{not
H}_{0}.$ (1.1)

In particular, the prominent examples of symmetry mentioned earlier correspond
to the following groups $\mathcal{G}$.

  1. (a)

Central symmetry: Here $\mathcal{G}=\\{I,-I\\}$, where $I\equiv I_{p}$ is the
identity matrix of order $p$.

Central symmetry (or “reflective” or “diagonal” or “simple” or “antipodal”
symmetry [91, 13, 80, 83, 32]) arises naturally from paired data (see e.g.,
[40, Section 5]): Given a sample of $n$ pairs
$\\{(\mathbf{Y}_{i},\mathbf{Z}_{i})\\}_{i=1}^{n}$, it is often of interest to
study the differences $\mathbf{X}_{i}:=\mathbf{Y}_{i}-\mathbf{Z}_{i}$,
$i=1,\ldots,n$. Under the null hypothesis that $\mathbf{Y}_{i}$ and
$\mathbf{Z}_{i}$ are exchangeable, $\mathbf{X}_{i}$ will have a centrally
symmetric distribution [67, 40]. Extensions of central symmetry include
angular symmetry [86], which corresponds to $\frac{{\bf X}}{\|{\bf
X}\|}\stackrel{{\scriptstyle d}}{{=}}-\frac{{\bf X}}{\|{\bf X}\|}$; here
$\|\cdot\|$ denotes the usual Euclidean norm.

  2. (b)

Sign symmetry: Here $\mathcal{G}=\\{Q\in\mathbb{R}^{p\times p}:Q{\rm\ is\ a\
diagonal\ orthogonal\ matrix}\\}$; thus each diagonal entry of $Q$ is $\pm 1$.

Sign symmetry is also called orthant symmetry or marginal symmetry [31, 84,
83]. Such sign-change invariance is often assumed in the location-scatter
model, which provides tools for robust estimation of the regular mean vector
and covariance matrix [83]; also see the book by Puri and Sen [85] which gives
a comprehensive account of multivariate analysis methods based on marginal
signs and ranks.

  3. (c)

Spherical symmetry: Here $\mathcal{G}={\rm O}(p)$.

Spherically symmetric distributions are natural generalizations of the
Gaussian distribution, and have drawn substantial interest [33, 2]. It is also
known that many estimation results are robust when departures from the
normality assumption are in the direction of the spherically symmetric family,
with applications in the general linear model [62, 10, 36], sufficient
dimension reduction [69, 105], etc. Tests for spherical symmetry have been
considered in [6, 39, 52, 2].

Clearly, other subgroups of ${\rm O}(p)$, such as the signed permutation
group111The signed permutation group consists of all matrices having exactly
one nonzero entry in each row and exactly one nonzero entry in each column,
and the nonzero entries are either 1 or $-1$. [35], also correspond to a
notion of symmetry. The $\mathcal{G}$-symmetric model described by ${\rm
H}_{0}$ in (1.1) is motivated by the work of Beran and Millar, [9], and it
provides a general framework for testing many notions of multivariate
symmetries in a unified fashion.

To develop distribution-free tests for the general hypothesis of
$\mathcal{G}$-symmetry in (1.1) we first observe (in Section 2) that the one-
dimensional signs, ranks and signed-ranks — which are the building blocks for
distribution-free inference in the univariate setting — can be obtained as
solutions to an optimization problem which can be generalized to multi-
dimensions. This resulting optimization is an optimal transport (OT) problem
[100, 101] with a specific cost function that can be solved efficiently (see
Section 2.3).

Specifically, given observations $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ from
$\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$ and (known) candidate rank
vectors $\mathbf{h}_{1},\ldots,\mathbf{h}_{n}\in\mathbb{R}^{p}$, the
generalized sign $S_{n}(\mathbf{X}_{i})$ is defined as an element of
$\mathcal{G}$ and the collection of generalized ranks
$(R_{n}(\mathbf{X}_{1}),\ldots,R_{n}(\mathbf{X}_{n}))$ is a permutation of
$(\mathbf{h}_{1},\ldots,\mathbf{h}_{n})$; see Section 2.2 for the formal
definitions. Our approach leads to notions of ($\mathcal{G}$-symmetric
specific) generalized multivariate signs, ranks and signed-ranks that are also
distribution-free (under ${\rm H}_{0}$). In the one-dimensional case, if
$(\mathbf{h}_{1},\ldots,\mathbf{h}_{n})$ is chosen as
$(\frac{1}{n},\ldots,\frac{n}{n})$, then we recover the usual notion of signs
and ranks. Further, mirroring the one-dimensional setting, we define the
generalized sign test statistic as $\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})$, and
the statistic for the generalized Wilcoxon signed-rank (GWSR) test will be
defined as $\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})$; see
Section 3 for the details.

Over the past few years, multivariate ranks [43, 27], defined via the theory
of OT, have been effectively used to design distribution-free testing
procedures in two-sample problems [15, 27], for independence testing [27, 95,
96, 94], in multivariate linear models [44, 45], for directional data [46],
etc. Our framework, although inspired by these recent developments, is novel
and more general and includes the usual notion of OT based multivariate ranks
[43, 27] as a special case. We achieve this by considering a more general cost
function in the OT problem that naturally arises from the hypothesis testing
problem (1.1) and reduces to the usual squared Euclidean distance cost when
$\mathcal{G}$ is the trivial group $\\{I\\}$. Further, in Section 2.4, using
the theory of OT, we provide characterizations of the population analogues of
the empirical generalized signs, ranks and signed-ranks (see Theorem 2.1).
Moreover, we show the consistency of these sample estimates to their
population analogues in Theorem 2.2. Our treatment does not need any moment
assumptions on the data distribution $\mu$ as we appeal to geometric
characterizations of OT (as pioneered by Robert McCann [73]).

Among the important reasons for the popularity of the classical Wilcoxon
signed-rank test are the striking results of Hodges and Lehmann, [57] and
Chernoff and Savage, [22], where the authors show that the asymptotic (Pitman)
relative efficiency of Wilcoxon’s test with respect to (w.r.t.) the one-sample
Student’s $t$-test, under location-shift alternatives, never falls below 0.864
(with the identity score function) and 1 (with the Gaussian score function)
respectively, despite the former being exactly distribution-free for all
sample sizes. Motivated by these results, we study in Section 3.4 the
(asymptotic relative) efficiency, in the sense of Pitman [see e.g., 99,
Section 14.3], of our proposed multivariate generalization of the Wilcoxon
signed-rank test compared to Hotelling’s $T^{2}$ test — the multivariate
generalization of the usual $t$-test. In particular, we show that the proposed
GWSR test satisfies Hodges-Lehmann and Chernoff-Savage-type efficiency lower
bounds over natural sub-families of multivariate distributions, despite being
entirely agnostic to the underlying data generating mechanism; see Theorems
3.5-3.7.

Let us now briefly highlight the main results in this paper.

  1. 1.

Distribution-freeness: under ${\rm H}_{0}$,

    1. (a)

$S_{n}(\mathbf{X}_{1}),\ldots,S_{n}(\mathbf{X}_{n})$ are i.i.d. following the
uniform distribution over $\mathcal{G}$.

    2. (b)

The ranks $(R_{n}(\mathbf{X}_{1}),\ldots,R_{n}(\mathbf{X}_{n}))$ are uniformly
distributed over all $n!$ permutations of
$(\mathbf{h}_{1},\ldots,\mathbf{h}_{n})$; see 2.2.

  2. 2.

Independence: $(S_{n}(\mathbf{X}_{1}),\ldots,S_{n}(\mathbf{X}_{n}))$ is
independent of $(R_{n}(\mathbf{X}_{1}),\ldots,R_{n}(\mathbf{X}_{n}))$ under
$\textrm{H}_{0}$ (2.2).

  3. 3.

Asymptotic normality: Both
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})$ and
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})$
converge in distribution to multivariate normal distributions, under ${\rm
H}_{0}$ (Theorem 3.1).

  4. 4.

Relative efficiency against location shift alternatives for suitable sub-
classes of multivariate distributions (e.g., product distributions,
elliptically symmetric distributions, etc.):

    1. (a)

(Hodges-Lehmann phenomenon [57]) The relative efficiency of the GWSR test
w.r.t. Hotelling’s $T^{2}$ test never falls below 0.864, if
$\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$ and $\mathcal{G}$ determine a uniform
effective reference distribution (see Theorem 3.5 for the precise statement).

    2. (b)

(Chernoff-Savage phenomenon [22]) The relative efficiency of the GWSR test
against Hotelling’s $T^{2}$ test is lower bounded by 1, if
$\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$ and $\mathcal{G}$ determine a Gaussian
effective reference distribution (see Theorems 3.5-3.7).

  5. 5.

Efficiency (against location shift alternatives): The family of score
transformed GWSR tests (with statistic
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})J(R_{n}(\mathbf{X}_{i}))$
for some score function $J:\mathbb{R}^{p}\to\mathbb{R}^{p}$) contains a
locally asymptotically optimal test (see Theorem 3.9), i.e., a test which has
maximum (asymptotic) local power among all possible tests with a fixed Type I
error. This is in parallel with the analogous result in univariate setting,
cf. Van der Vaart, [99, Corollary 15.10].

We say that $\mathbf{X}$ is $\mathcal{G}$-symmetric around
${\bm{\theta}}\in\mathbb{R}^{p}$ (usually referred to as the center of
$\mathcal{G}$-symmetry) if
$(\mathbf{X}-{\bm{\theta}})\overset{d}{=}Q(\mathbf{X}-{\bm{\theta}})$, for all
$Q\in\mathcal{G}$. Our framework can be immediately extended to test the
hypothesis of $\mathcal{G}$-symmetry around any fixed and known
${\bm{\theta}}$. To test symmetry around a general center
${\bm{\theta}}\in\mathbb{R}^{p}$, one may just translate the data
$\\{\mathbf{X}_{i}\\}_{i=1}^{n}$ to
$\\{\mathbf{X}_{i}-{\bm{\theta}}\\}_{i=1}^{n}$, thereby reducing the problem
to (1.1). Further, if ${\bm{\theta}}$ is unknown, our proposed testing
framework can be used to construct distribution-free confidence sets for
${\bm{\theta}}$ (see Section 5). Further, in Section 4 we demonstrate the
superior performance of our proposed tests (especially for location shift
models) via simulation studies, illustrate the implications of our efficiency
results for data sets with small to moderate sample sizes, and compare our
proposals with existing methods that test for different notions of symmetry.

From the above properties, we believe that our generalized multivariate signs
and ranks (defined via OT) are natural extensions of the classical concepts.
As far as we are aware, our work is the first to propose notions of
multivariate distribution-free signs and ranks (relative to any group
$\mathcal{G}$) and develop distribution-free testing for (1.1) in multi-
dimensions with the above efficiency lower bounds, analogous to the classical
Wilcoxon’s signed-rank test.

###  1.1 Literature Review

Multivariate symmetry is an important concept in nonparametric statistics
[106]. Testing for multivariate symmetry has been studied extensively in the
literature, see, e.g., Baringhaus, [6], Diks and Tong, [28] and Serfling,
[93].

Extensions of sign in multi-dimension. One of the earliest and most popular
tests for symmetry is possibly the sign test [4]. Since then, several notions
of multivariate signs have been proposed in the literature. The component-wise
multivariate sign, which is a popular generalization, dates back to Bennett,
[8], Chatterjee, [20]. The asymptotic normality and Pitman efficiency of the
component-wise sign test against Hotelling’s $T^{2}$ was derived by Bickel,
[12]. Component-wise ranks and signed-ranks have also been used for testing
[85, 59]. One drawback of the component-wise sign is that its distribution-
freeness requires $\mathbf{X}$ to be sign symmetric (it is not distribution-
free under central symmetry). We will show that our generalized sign (see
(2.4)) reduces to the component-wise sign when $\mathcal{G}$ is the group
corresponding to sign symmetry (see Remark 2.4).

A multivariate generalization of the sign test based on interdirections was
proposed in Randles, [86], generalizing the two-sided univariate sign test and
Blumen’s bivariate sign test [14]. This test statistic is invariant under
nonsingular linear transformations and it has a finite-sample distribution-
free property over distributions with elliptical directions [86]. Chaudhuri
and Sengupta, [21] expressed Randles’s statistic in terms of a data-dependent
sign, and further unified a number of previous affine invariant generalized
sign tests [56, 102, 1]. Such transformation re-transformation technique was
later investigated to define affine equivariant multivariate median and
quantiles [18, 19, 17, 92], and affine invariant multivariate sign test [87].
Locally asymptotically maximin tests based on interdirections were derived in
Hallin and Paindaveine, [47] under elliptical symmetry.

Möttönen and Oja, [75] defined the multivariate sign as the unit vector in the
direction of a given point, which is called the spatial sign. The Pitman
efficiencies of the corresponding tests and a generalization of the Hodges-
Lehmann estimator were further analyzed in Möttönen et al., [77, 76]. However,
none of the above notions of signs are distribution-free beyond spherical
symmetry.

Hallin et al., [43] proposed a novel center-outward definition of ranks and
signs based on measure transportation with attractive properties, similar to
our framework with $\mathcal{G}=\\{I\\}$. However, their center-outward signs
differ from the classical signs and our proposal in multiple ways; e.g., the
center-outward signs are not i.i.d. uniform over $\\{\pm 1\\}$ for symmetric
distributions in one dimension.

Tests for central symmetry. Going beyond the sign test, there are tests for
central symmetry which are consistent against all alternatives. In the
univariate setting, McWilliams, [74] presented a simple test based on a runs
statistic. Dyckerhoff et al., [29] extended McWilliams’s procedure to test for
bivariate central symmetry using the notion of data depth. In Einmahl and Gan,
[32], omnibus tests for central symmetry of a bivariate probability
distribution are proposed that are asymptotically distribution-free. Another
class of approaches explores the properties of the characteristic function of
a centrally symmetric distribution and uses the empirical characteristic
function to construct test statistics, including both in the univariate
setting [24, 34] and in multi-dimensional scenarios [39, 51, 53, 80, 54].
Affine invariant tests for central symmetry are proposed in Henze et al., [53]
using kernel density estimators; also see [54]. However, these tests are in
general not asymptotically distribution-free, and the limiting distributions
need to be estimated from the data for inference [80, 54].

Tests for spherical symmetry. A direct extension of McWilliams’s procedure for
testing spherical symmetry in multi-dimension is given in Marden, [71, Section
2.4], which has a simple asymptotic $\chi^{2}_{1}$ distribution under the
null. In Baringhaus, [6], rotationally invariant tests based on test
statistics of the von Mises type are proposed, which give rise to tests that
are consistent against all alternatives. The characteristic function has also
been used for testing spherical symmetry in multi-dimensional scenarios [52,
39]. Kariya and Eaton, [64] consider two classes of alternatives to test
spherical symmetry, and provide uniformly most powerful tests. A new testing
procedure based on the fact that $\mathbf{X}$ is spherically symmetric if and
only if
$\mathbb{E}[\mathbf{u}^{\top}\mathbf{X}|\mathbf{v}^{\top}\mathbf{X}]=0$ for
any two perpendicular vectors $\mathbf{u}$ and $\mathbf{v}$ is given in
Albisetti et al., [2], with its rejection region being found via bootstrap.

Tests for sign symmetry. The analysis of the classical Hotelling’s $T^{2}$
test has also been extended to sign symmetric distributions by Efron, [31],
Pinelis, [84], etc. In Oja, [83], a book-length treatment on tests based on
spatial signs and ranks is given; sign-change versions of the spatial sign
test and spatial signed-rank test are proposed when the observations are sign
symmetric. These tests also have good efficiency properties with respect to
Hotelling’s $T^{2}$ in the multivariate $t$-distribution settings [83, Chapter
8]. However, Hodges-Lehmann [57] and Chernoff-Savage [22] type lower bounds
are not available for these methods.

###  1.2 Organization of the Paper

In Section 2, we define multivariate signs, ranks and signed-ranks, and
provide their population analogues. In Section 3, we define multivariate
analogues of the sign test and the Wilcoxon signed-rank test. We further
investigate the statistical properties of these proposed tests such as
distribution-freeness, asymptotic normality, consistency, and efficiency.
Finite sample performance of the proposed tests and comparisons with competing
methods are given in Section 4. A distribution-free confidence set for the
unknown center of symmetry of a $\mathcal{G}$-symmetric distribution is
constructed in Section 5. All proofs of our results and further discussions
are relegated to Appendices A-C.

##  2 Multivariate Generalized Signs, Ranks and Signed-ranks

###  2.1 Preliminaries

Let ${\rm O}(p)$, the orthogonal group in dimension $p$, denote the group of
all $\mathbb{R}^{p\times p}$ orthogonal matrices under the group operation of
matrix multiplication. Let $\mathcal{G}$ be a compact subgroup of the
orthogonal group ${\rm O}(p)$. In particular, $\mathcal{G}$ satisfies the
following group axioms: (i) $I\in\mathcal{G}$; here $I\equiv I_{p}$ is the
identity matrix of order $p$; (ii) if $Q\in\mathcal{G}$, then
$Q^{-1}\in\mathcal{G}$; (iii) if $Q_{1}$, $Q_{2}\in\mathcal{G}$, then
$Q_{1}Q_{2}\in\mathcal{G}$; (iv) if $Q_{1}$, $Q_{2}$, $Q_{3}\in\mathcal{G}$,
then $(Q_{1}Q_{2})Q_{3}=Q_{1}(Q_{2}Q_{3})$. The compactness assumption ensures
that there exists a unique uniform distribution (also called the Haar measure)
over $\mathcal{G}$; see e.g., Eaton, [30].

######  Definition 2.1 (Orbit).

The orbit of an element $\mathbf{x}\in\mathbb{R}^{p}$ is the set of elements
(in $\mathbb{R}^{p}$) to which $\mathbf{x}$ can be moved by the elements of
$\mathcal{G}$, i.e., $\\{Q\mathbf{x}:Q\in\mathcal{G}\\}$.

Note that, if $\mathbf{y}$ is in the orbit of $\mathbf{x}$, then $\mathbf{x}$
is also in the orbit of $\mathbf{y}$. Thus, $\mathbb{R}^{p}$ can be
partitioned into a disjoint union of orbits. Let $B\subset\mathbb{R}^{p}$. We
define the set

$\mathcal{G}B:=\\{Q\mathbf{x}:Q\in\mathcal{G},\mathbf{x}\in
B\\}\subset\mathbb{R}^{p}$ (2.1)

as the image of the group action of $\mathcal{G}$ on $B$.

######  Definition 2.2 (Group action).

A (left) group action of $\mathcal{G}$ on $\mathcal{G}B$ is a function
$\cdot:\mathcal{G}\times\mathcal{G}B\to\mathcal{G}B$ which satisfies the
following two axioms: (i) $I\cdot\mathbf{x}=\mathbf{x}$, and (ii)
$Q_{1}\cdot(Q_{2}\cdot\mathbf{x})=(Q_{1}Q_{2})\cdot\mathbf{x}$, for all
$\mathbf{x}\in\mathcal{G}B$, and $Q_{1},Q_{2}\in\mathcal{G}$. In our setting
we take the group action $\cdot$ to be the usual matrix multiplication
operation.

######  Definition 2.3 (Free group action).

We say that $\mathcal{G}$ acts freely on $\mathcal{G}B$ if for $\mathbf{y}\in
B$ and $Q\in\mathcal{G}$, $Q\mathbf{y}=\mathbf{y}$ implies $Q=I$ (i.e., for
any $\mathbf{x}$ in $\mathcal{G}B$, only the identity in $\mathcal{G}$ leaves
$\mathbf{x}$ fixed).

Intuitively, a free group action helps identify the unique element in
$\mathcal{G}$ that maps a point to another point (i.e.,
$Q_{1}\mathbf{x}=Q_{2}\mathbf{x}$ for $Q_{1},Q_{2}\in\mathcal{G}$ and
$\mathbf{x}\in\mathcal{G}B$ implies that $Q_{1}=Q_{2}$). Let $\mathcal{P}_{\rm
ac}(\mathbb{R}^{p})$ denote the class of all Borel probability measures on
$\mathbb{R}^{p}$ with a Lebesgue density. We say that a probability measure
$\mu$ is concentrated on a set $B$ if $\mu(B)=1$. If $\mu$ is a Borel measure,
the support of $\mu$ is the smallest closed set on which $\mu$ is
concentrated.

######  Definition 2.4 (Push forward distribution).

Let $F:\mathbb{R}^{p}\to\mathbb{R}^{k}$ (for $k\geq 1$) be a function and
$\mu$ be a distribution on $\mathbb{R}^{p}$. The push forward distribution
$F\\#\mu$ is defined as the distribution of $F(\mathbf{X})$, where
$\mathbf{X}\sim\mu$.

######  Definition 2.5 (Cyclical monotonicity).

Let $c:\mathbb{R}^{p}\times\mathbb{R}^{p}\to(-\infty,+\infty]$ be a function.
A subset $\Gamma\subset\mathbb{R}^{p}\times\mathbb{R}^{p}$ is said to be
$c$-cyclically monotone if for any $N\in\mathbb{N}$ and any family
$(\mathbf{x}_{1},\mathbf{y}_{1}),\ldots,(\mathbf{x}_{N},\mathbf{y}_{N})$ of
points in $\Gamma$, the following inequality holds:

$\sum_{i=1}^{N}c(\mathbf{x}_{i},\mathbf{y}_{i})\leq\sum_{i=1}^{N}c(\mathbf{x}_{i},\mathbf{y}_{i+1}),$

with the convention $\mathbf{y}_{N+1}=\mathbf{y}_{1}$.

###  2.2 Connection to Optimal Transport

For a random vector ${\bf X}\sim\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$
and a compact subgroup $\mathcal{G}$ of the orthogonal group ${\rm O}(p)$, we
are interested in distribution-free testing of the hypothesis of
$\mathcal{G}$-symmetry (1.1). In this subsection we define new distribution-
free (in finite samples) multivariate generalizations of signs and ranks that
will form the building blocks for distribution-free inference. Further, these
generalizations of signs and ranks — which reduce to the usual classical
notions of sign and rank when $p=1$ and $\mathcal{G}=\\{I,-I\\}$ — adapt to
different notions of multivariate symmetry, and share many properties similar
to their classical counterparts. We borrow ideas from the theory of optimal
transport (OT) to define our multivariate generalizations. Although our
approach is inspired by the notion of multivariate ranks defined via OT [43,
27], our framework is novel and more general.

Let $\nu$ be some known and fixed Borel measure on $\mathbb{R}^{p}$, which
will be referred to as the reference distribution; see Remark 3.3 for concrete
examples. For each $n$, we consider a quasi-Monte Carlo [48, 66, 81] or Monte
Carlo [50, 88] sequence
$\\{\mathbf{h}_{n,j}\in\mathbb{R}^{p}:\,j=1,2,\ldots,n\\}$ assumed to be known
and fixed, such that the empirical distribution over
$\\{\mathbf{h}_{n,j}\\}_{j=1}^{n}$, denoted by $\nu_{n}$, converges in
distribution to $\nu$ as $n\to\infty$. So $\nu_{n}$ provides a discretization
of $\nu$. For notational brevity, we hide $n$ and write
$\mathbf{h}_{j}\equiv\mathbf{h}_{n,j}$. For instance,
$\\{\mathbf{h}_{j}\\}_{j=1}^{n}$ can be taken as $n$ i.i.d. observations from
$\nu$ and then fixed. If $\nu={\rm Uniform}(0,1)^{p}$, we can also let
$\\{\mathbf{h}_{j}\\}_{j=1}^{n}$ be the first $n$ points in the Halton
sequence [48]; other choices are also available (see e.g., Deb and Sen, [27,
Appendix D.3]). As we will see later, to recover the classical one-dimensional
sign and signed-rank, one can take $\nu={\rm Uniform}(0,1)$ and
$\mathbf{h}_{j}=\frac{j}{n}$, for $j=1,2,\ldots,n$.

Let $\mathcal{S}_{n}$ be the set of all $n!$ permutations of
$\\{1,\ldots,n\\}$. Consider the following optimization problem:

$\min\left\\{\sum_{i=1}^{n}\lVert
Q_{i}^{\top}\mathbf{X}_{\sigma(i)}-\mathbf{h}_{i}\rVert^{2}:Q_{i}\in\mathcal{G},\sigma\in\mathcal{S}_{n}\right\\}.$
(2.2)

Let $(\\{\widehat{Q}_{i}\\}_{i=1}^{n},\widehat{\sigma})$ be a minimizer of
(2.2). In fact, under mild assumptions, $\widehat{\sigma}$ is actually a.s.
unique, and $\\{\widehat{Q}_{i}\\}_{i=1}^{n}$ is also a.s. unique if
$\mathcal{G}$ induces a free group action (see 2.1). Define the cost function

$c(\mathbf{x},\mathbf{h}):=\min_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2},\qquad\mbox{for}\;\;\mathbf{x},\mathbf{h}\in\mathbb{R}^{p},$
(2.3)

so that (2.2) reduces to
$\min_{\sigma\in\mathcal{S}_{n}}\sum_{i=1}^{n}c(\mathbf{X}_{\sigma(i)},\mathbf{h}_{i})$
— the optimal transport (OT) problem from
$\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}$ to
$\\{\mathbf{h}_{1},\ldots,\mathbf{h}_{n}\\}$ under the cost function
$c(\cdot,\cdot)$. Observe that the cost function satisfies
$c(\mathbf{x},\mathbf{h})=c(\mathbf{x},Q\mathbf{h})=c(Q\mathbf{x},\mathbf{h})$
for any $Q\in\mathcal{G}$, i.e., it is invariant under transformations in
$\mathcal{G}$.

Define the generalized sign
$S_{n}:\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}\to\mathcal{G}$ and
generalized rank
$R_{n}:\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}\to\\{\mathbf{h}_{1},\ldots,\mathbf{h}_{n}\\}$
as:

$S_{n}(\mathbf{X}_{i})=\widehat{Q}_{\widehat{\sigma}^{-1}(i)},\qquad\mbox{and}\qquad
R_{n}(\mathbf{X}_{i})=\mathbf{h}_{\widehat{\sigma}^{-1}(i)},\qquad\mbox{for}\;\;i=1,\ldots,n.$
(2.4)

The generalized signed-rank of the $i$-th observation $\mathbf{X}_{i}$ is then
defined as $S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})$.

######  Remark 2.1 (When $p=1$).

Note that when $p=1$ and $\mathcal{G}=\\{1,-1\\}$, the generalized signs and
ranks (in (2.4)) reduce to the classical one-dimensional concepts. In fact,
problem (2.2) directly yields the usual signs and ranks in this case. To see
this observe that, for $x\in\mathbb{R}$ and $h\in[0,1]$, from (2.3),
$c(x,h)=\min\\{(x-h)^{2},(-x-h)^{2}\\}$, which (due to the linear ordering of
the real line) reduces to $c(x,h)=(|x|-h)^{2}$. It is now easy to see that
(2.2), with $h_{j}=\frac{j}{n}$ for all $j$, simplifies to

$\min_{\sigma\in\mathcal{S}_{n}}\sum_{i=1}^{n}\left(|X_{i}|-\frac{\sigma(j)}{n}\right)^{2}=\max_{\sigma\in\mathcal{S}_{n}}\sum_{i=1}^{n}|X_{i}|\frac{\sigma(j)}{n},$

which, by the rearrangement inequality (see e.g., [49, Theorem 368]), yields
that $\widehat{\sigma}(i)$ will equal the usual (absolute) rank of $|X_{i}|$,
for $i=1,\ldots,n$. Thus, $X_{i}$ with the $j$-th largest absolute value will
be mapped to $j/n$, and the sign of $X_{i}$ will be 1 if $X_{i}\geq 0$ and
$-1$ if $X_{i}<0$.

When $p>1$, due to the absence of a canonical ordering in $\mathbb{R}^{p}$,
optimization problem (2.2) does not have an explicit solution. However, the
approach is similar; optimization problem (2.2) assigns the data points
$\\{\mathbf{X}_{i}\\}_{i=1}^{n}$ to the canonical multivariate rank vectors
$\\{\mathbf{h}_{j}\\}_{j=1}^{n}$ in a bijective fashion such that the
objective function in (2.2) is minimized. To develop some intuition for the
generalized signs and signed-ranks (as in (2.4)), maybe it is easier to
express the cost function as
$c(\mathbf{x},\mathbf{h})=\min_{Q\in\mathcal{G}}\|\mathbf{x}-Q\mathbf{h}\|^{2}$.
Thus, if $R_{n}(\mathbf{X}_{i})=\mathbf{h}_{j}$, then
$c(\mathbf{X}_{i},\mathbf{h}_{j})=\min_{Q\in\mathcal{G}}\|\mathbf{X}_{i}-Q\mathbf{h}_{j}\|^{2}$
and hence the signed-rank $S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})\equiv
S_{n}(\mathbf{X}_{i})\mathbf{h}_{j}$ is the closest element (in Euclidean
norm) to $\mathbf{X}_{i}$ in the orbit of $\mathbf{h}_{j}$. Moreover,
$S_{n}(\mathbf{X}_{i})$, the sign of $\mathbf{X}_{i}$, is the argmin of the
cost $\|\mathbf{X}_{i}-Q\mathbf{h}_{j}\|^{2}$ over $\mathcal{G}$ — it is the
orthogonal matrix in $\mathcal{G}$ that transforms the rank vector
$R_{n}(\mathbf{X}_{i})\equiv\mathbf{h}_{j}$ to bring it closest to
$\mathbf{X}_{i}$.

Figure 1: Left: The data points $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ (marked
by “ $\bullet$”) with $n=7$ drawn from $N(\mathbf{0}_{2},4I_{2})$. The rank
vectors $\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$ (marked by “+”) are taken as
$n$ i.i.d. draws from the distribution of $(|Z_{1}|,Z_{2})$, where
$Z_{1},Z_{2}$ are i.i.d. from $N(0,1)$. Right (classical signs and ranks):
$n=3$ data points (marked by “ $\bullet$”) are drawn from $N(0,4)$. The ranks
(marked by “+”) are $h_{j}=\frac{j}{n}$, $j=1,\ldots,n$. The squared length of
each solid line is the cost of transporting each data point. The dashed
(dotted) lines indicate that $\mathbf{X}_{i}\mapsto-\mathbf{X}_{i}$ (i.e.,
$S_{n}(\mathbf{X}_{i})=-I$) before being transported to the
$\mathbf{h}_{j}$’s.

Figure 1 illustrates this for $p=1$ and $p=2$ with $\mathcal{G}=\\{-I,I\\}$.
The left panel of Figure 1 shows an example consisting of $n=7$ data points
$\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ (marked by “ $\bullet$”) from
$N(\mathbf{0}_{2},4I_{2})$ and their generalized ranks when
$\mathcal{G}=\\{I,-I\\}$ (corresponding to central symmetry). The rank vectors
$\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$ (marked by “+”) are taken as $n$ i.i.d.
draws from the distribution of $(|Z_{1}|,Z_{2})$, where $Z_{1},Z_{2}$ are
i.i.d. from $N(0,1)$. The squared length of each solid line in Figure 1 is the
cost of transporting each data point, i.e.,
$c(\mathbf{X}_{i},R_{n}(\mathbf{X}_{i}))\equiv\|\mathbf{X}_{i}-S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})\|^{2}$,
for $i=1,\ldots,n$. For some data points $\mathbf{X}_{i}$, $\mathbf{X}_{i}$ is
first transported to $-\mathbf{X}_{i}$ without incurring any cost if
$\min_{Q\in\\{-I,I\\}}\|Q^{\top}\mathbf{X}_{i}-R_{n}(\mathbf{X}_{i})\|^{2}=\|-\mathbf{X}_{i}-R_{n}(\mathbf{X}_{i})\|^{2}$
(as shown by the dashed line). In such a case, $S_{n}(\mathbf{X}_{i})$, the
sign of $\mathbf{X}_{i}$, is $-I$. If directly mapping $\mathbf{X}_{i}$ to
$R_{n}(\mathbf{X}_{i})$ has a smaller squared distance compared to that
between $-\mathbf{X}_{i}$ and $R_{n}(\mathbf{X}_{i})$, then
$S_{n}(\mathbf{X}_{i})=I$. When $p=1$ (see the right plot in Figure 1),
similar observations can be made. It is just that in this case optimization
problem (2.2) can be explicitly solved (as shown in Remark 2.1).

The following proposition (proved in Section B.1) shows that under mild
assumptions, the generalized ranks and signed-ranks are a.s. unique. Although
the generalized signs may not be unique, they are a.s. unique when the group
action of $\mathcal{G}$ is free.

######  Proposition 2.1 (Uniqueness of sample ranks, signs, and signed-ranks).

Assume that ${\bf X}\sim\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$, and
suppose that no two $\mathbf{h}_{j}$’s lie on a same orbit of $\mathcal{G}$.

  1. 1.

Then the multivariate rank $R_{n}(\cdot)$ and signed-rank
$S_{n}(\cdot)R_{n}(\cdot)$ are a.s. unique.

  2. 2.

If $\mathbf{h}_{1},\ldots,\mathbf{h}_{n}\in B\subset\mathbb{R}^{p}$, and
$\mathcal{G}$ acts freely on $\mathcal{G}B$ (see (2.1)), then $S_{n}(\cdot)$
is a.s. unique.

A free group action (see Definition 2.3) is easily available for central
symmetry and sign symmetry by taking, for example,
$B=(0,\infty)\times\mathbb{R}^{p-1}$ and $B=(0,\infty)^{p}$ respectively.
Thus, for these groups the generalized sign $S_{n}(\cdot)$ is unique; see
Figure 2(a) (central symmetry) and Figure 2(b) (sign symmetry).

However, for spherical symmetry, we do not have a free group action when
$p>1$: In Figure 2(c) (spherical symmetry), $\mathcal{G}={\rm O}(2)$ and the
generalized sign is not unique — rotation by 90 degrees and reflection along
the line $y=x$ both map the rank vector to the signed-rank. However, the
signed-rank is unique — it is the point in
$\\{\mathbf{x}\in\mathbb{R}^{2}:\|\mathbf{x}\|=2\\}$ (as the rank vector
$\mathbf{h}$, indicated by “+” in Figure 2(c), satisfies $\|\mathbf{h}\|=2$)
that is closest to the data point.

######  Remark 2.2 (When the minimizer of (2.2) is not unique).

In cases where the minimizer of (2.2) is not unique, perform a random draw
from all possible minimizers as follows: If
$\sigma_{1},\ldots,\sigma_{k}\in\mathcal{S}_{n}$ all attain the minimal cost
$\sum_{i=1}^{n}c(\mathbf{X}_{\sigma(i)},\mathbf{h}_{i})$, set
$\widehat{\sigma}$ to be $\sigma_{j}$ with probability
$\mathbb{P}(\widehat{\sigma}=\sigma_{j})=\frac{1}{k}$, $j=1,\ldots,k$. After
determining $\widehat{\sigma}$, find the minimizer
$\widehat{Q}_{i}:=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q_{i}^{\top}\mathbf{X}_{\widehat{\sigma}(i)}-\mathbf{h}_{i}\|^{2}$.
If this minimizer is not unique, introduce another random variable/vector
(independent of everything else) to choose $\widehat{Q}_{i}$ from the uniform
distribution over $\mathcal{G}$ conditioning on that it minimizes $\lVert
Q_{i}^{\top}\mathbf{X}_{\widehat{\sigma}(i)}-\mathbf{h}_{i}\rVert^{2}$; see
Section 3.3 on how to sample from this conditional distribution.

((a)) $\mathcal{G}$: central symmetry.

((b)) $\mathcal{G}$: sign symmetry.

((c)) $\mathcal{G}$: spherical symmetry.

Figure 2: Illustration of a data point (marked by “ $\bullet$”), its rank
vector (marked by “+”), and its signed-rank (marked by “$\star$”). The signed-
rank is the unique point in the orbit of the rank vector that is closest to
the data point. For (c), where $\mathcal{G}$ corresponds to spherical
symmetry, the sign is not unique as rotation by 90 degrees and reflection
along the line $y=x$ both map the rank vector to the signed-rank.

The following proposition (see Section B.2 for a proof) shows that the
generalized signs and signed-ranks, possibly chosen via randomization as
described in Remark 2.2 if they are not unique, extend naturally the
properties of their classical counterparts (when $p=1$). In particular, under
the null hypothesis of $\mathcal{G}$-symmetry, the generalized signs and
generalized ranks are independent, and any statistic based on them is exactly
distribution-free in finite samples.

######  Proposition 2.2.

Consider the generalized signs, ranks and signed-ranks as defined in (2.4). If
they are not unique (cf. Proposition 2.1), we assume that they are chosen via
randomization as described in Remark 2.2. Then, the following properties hold:

  1. 1.

The generalized rank $(R_{n}(\mathbf{X}_{1}),\ldots,R_{n}(\mathbf{X}_{n}))$ is
independent of any order statistics222An order statistic is an un-ordered
version of the data $\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}$ [43]. of
$\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}$, and is uniformly distributed
over the set of all $n!$ permutations of
$(\mathbf{h}_{1},\ldots,\mathbf{h}_{n})$.

  2. 2.

$(R_{n}(\mathbf{X}_{1}),\ldots,R_{n}(\mathbf{X}_{n}))$ and the generalized
signs $(S_{n}(\mathbf{X}_{1}),\ldots,S_{n}(\mathbf{X}_{n}))$ are independent
under ${\rm H}_{0}:\mathbf{X}\overset{d}{=}Q\mathbf{X}$, for all
$Q\,\in\mathcal{G}$.

  3. 3.

$S_{n}(\mathbf{X}_{1}),\ldots,S_{n}(\mathbf{X}_{n})$ are i.i.d. following the
uniform distribution over $\mathcal{G}$, under ${\rm H}_{0}$.

######  Remark 2.3 (Unsigned multivariate rank).

If $\mathcal{G}=\\{I\\}$, then $S_{n}(\mathbf{X}_{i})=I$ and
$R_{n}(\mathbf{X}_{i})$ reduces to the (unsigned) multivariate rank defined in
Deb and Sen, [27], and includes the center-outward ranks proposed in Hallin et
al., [43].

######  Remark 2.4 (Component sign).

For sign-symmetry, when $\nu_{n}$ is concentrated on $(0,\infty)^{p}$,
$S_{n}(\mathbf{X}_{i})$ is a diagonal matrix with diagonal entries being
exactly equal to the component signs of $\mathbf{X}_{i}$ [8, 12, 20, 85].

###  2.3 Computational Complexity

The optimization problem (2.2) can be solved efficiently. Recall that
$c(\mathbf{x},\mathbf{h})=\min_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}$
is the cost function (see (2.3)), which can be computed in $O(1)$ time if
$\mathcal{G}$ is finite. Then the ranks $R_{n}(\mathbf{X}_{i})$ can be found
by solving the assignment problem [78, 11] of
$\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ to
$\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$ under the cost $c(\cdot,\cdot)$, for
which algorithms with worst case complexity $O(n^{3})$ are available [63].
After obtaining the ranks,
$S_{n}(\mathbf{X}_{i})=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{X}_{i}-R_{n}(\mathbf{X}_{i})\|^{2}$
can be obtained (see the discussion in Section 3.3 and Remark 3.1 on how to
obtain this minimizer for the three main examples of symmetry considered in
this paper).

######  Remark 2.5 (Fast computation for spherical symmetry).

When $\mathcal{G}={\rm O}(p)$, the computation of the ranks (and the signed-
ranks) is even faster (i.e., in $O(n\log n)$ time) since
$c(\mathbf{x},\mathbf{h})=\min_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}=\min_{Q\in\mathcal{G}}\left\\{\|\mathbf{x}\|^{2}-2\mathbf{x}^{\top}Q\mathbf{h}+\|\mathbf{h}\|^{2}\right\\}=\big{(}\|\mathbf{x}\|-\|\mathbf{h}\|\big{)}^{2}$,
where the last step follows from the fact that there exists $Q\in\mathcal{G}$
that maps $\mathbf{h}$ to the direction of $\mathbf{x}$, e.g., a Householder
reflection [58]. Thus, if $\mathbf{X}_{i}$ has the $j$-th largest Euclidean
norm among $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ and
$\|\mathbf{h}_{1}\|<\ldots<\|\mathbf{h}_{n}\|$, then $\mathbf{X}_{i}$ will
have $\mathbf{h}_{j}$ as its rank. The signed-rank of $\mathbf{X}_{i}$ is
simply the vector in the direction of $\mathbf{X}_{i}$ with length
$\|R_{n}(\mathbf{X}_{i})\|$, i.e.,
$S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})=\|R_{n}(\mathbf{X}_{i})\|\frac{\mathbf{X}_{i}}{\|\mathbf{X}_{i}\|}$.

###  2.4 The Population Analogues

So far, we have defined the sample version of the multivariate generalized
signs and ranks, $S_{n}(\cdot)$ and $R_{n}(\cdot)$, via the optimization
problem (2.2). We will now show that, under finite second order moment
assumptions on $\mu$ and $\nu$, the sample rank map $R_{n}(\cdot)$ converges
to a population rank map $R(\cdot)$ such that $(\mathbf{X},R(\mathbf{X}))$
solves the following OT problem (aka Kantorovich relaxation [101]):

$\inf_{(\mathbf{X},\mathbf{H}):(\mathbf{X},\mathbf{H})\in\Pi(\mu,\nu)}\mathbb{E}\left[c(\mathbf{X},\mathbf{H})\right],\quad{\rm
with}\
c(\mathbf{x},\mathbf{h})=\min_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}.$
(2.5)

Here, $\Pi(\mu,\nu)$ consists of all joint Borel distributions
$(\mathbf{X},\mathbf{H})$ with marginals $\mathbf{X}\sim\mu$ and
$\mathbf{H}\sim\nu$. It can be seen that the optimization problem (2.5) is a
natural population analogue of (2.2).

To establish the convergence of $S_{n}(\cdot)$ and $R_{n}(\cdot)$, we need the
reference distribution $\nu$ and the group $\mathcal{G}$ to satisfy certain
natural compatibility assumptions. In particular, we assume that:

######  Assumption 1.

There exists a Borel set $B\subset\mathbb{R}^{p}$ with $\nu(B)=1$ such that
for any $\mathbf{x}\in\mathbb{R}^{p}$, the orbit
$\\{Q\mathbf{x}:Q\in\mathcal{G}\\}$ of $\mathbf{x}$ intersects $B$ at one
point at most.

This assumption requires that only one point at most should be taken as a rank
vector (which belongs to $B$) from any orbit of $\mathcal{G}$. This is because
transporting $\mathbf{x}$ to any point in an orbit of $\mathcal{G}$ has the
same cost due to the form of the cost function $c(\cdot,\cdot)$ (see (2.3)),
and there is no need to include two representative points from an orbit, which
may break the uniqueness of the population OT map.333Let
$\mu=\nu=N(\mathbf{0},I)$. Then $R(\mathbf{x})=\mathbf{x}$ and
$R(\mathbf{x})=-\mathbf{x}$ are both OT maps with 0 loss (see (2.5)), provided
$-I\in\mathcal{G}$.

To formally state our result, we introduce some notation.

######  Definition 2.6 (Quotient map $q$).

Under 1 we define the quotient map $q:\mathcal{G}B\to B$ by
$q(Q\mathbf{x})=\mathbf{x}$ for $\mathbf{x}\in B$ and $Q\in\mathcal{G}$.
Observe that 1 ensures that $q$ is well-defined.

Note that the above definition is slightly different from the usual way a
quotient map is defined, e.g., as a map that takes a point to an equivalence
class (i.e., an orbit here); see e.g., Munkres, [79, Chapter 2.22]. Instead,
our quotient map $q$ explicitly maps all points in the orbit
$\\{Q\mathbf{x}:Q\in\mathcal{G}\\}$ to a representative point $\mathbf{x}$ in
$B$.

((a)) $\mathcal{G}$: central symmetry.

((b)) $\mathcal{G}$: sign symmetry.

((c)) $\mathcal{G}$: spherical symmetry.

Figure 3: Illustrations of the quotient map $q(\cdot)$ for different groups
$\mathcal{G}$, when $p=2$. Here, each blue point (marked by “ $\bullet$”) is
mapped to a representative point in its orbit (marked by “+”) that lies in the
set $B$ (shaded in yellow).

Figure 3 shows examples of quotient maps. In Figure 3(a),
$B=(0,\infty)\times\mathbb{R}$ and $\mathcal{G}=\\{-I,I\\}$. Here
$q(\mathbf{x})=-\mathbf{x}$ if $x_{1}<0$ and $q(\mathbf{x})=\mathbf{x}$ if
$x_{1}>0$, where $\mathbf{x}=(x_{1},x_{2})$. Figure 3(b) shows the actions of
the quotient map $q$ on the three points $(-1,2)$, $(-1,-2)$ and $(1,-2)$, for
$\mathcal{G}$ being the group corresponding to sign symmetry with
$B=(0,\infty)^{2}$. All these three points are in the same orbit of
$\mathcal{G}$ and are mapped to the same point $(1,2)$. In this case,
$q(\mathbf{x})=(|x_{1}|,|x_{2}|)$. Figure 3(c) shows the quotient map for
$\mathcal{G}={\rm O}(2)$, corresponding to spherical symmetry, with
$B=(0,\infty)\times\\{0\\}$. In this case, the quotient map $q$ can be written
as $q(\mathbf{x})=(\|\mathbf{x}\|,0)$.

The following result (proved in Section B.4) states the existence and
uniqueness of the population rank map $R:\mathbb{R}^{p}\to\mathbb{R}^{p}$ such
that $({\rm identity},R)\\#\mu$ (i.e., the distribution of
$(\mathbf{X},R(\mathbf{X}))$ where $\mathbf{X}\sim\mu$) solves (2.5), if the
optimal cost is finite. Even if the optimal cost is infinite, $R(\cdot)$ can
be uniquely characterized using the notion of $c$-cyclical monotonicity (see
Definition 2.5).

######  Theorem 2.1 (Population rank and signed-rank maps).

Suppose 1 holds. Denote the distribution of $\mathbf{X}$ by
$\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$. Then there exists a $\mu$-a.e.
unique Borel measurable map $R:\mathbb{R}^{p}\to\mathbb{R}^{p}$ such that
$(\mathbf{X},R(\mathbf{X}))$ has the unique distribution in $\Pi(\mu,\nu)$
with a $c$-cyclically monotone support. Moreover, $R$ has the following
properties:

  1. (i)

Let $q$ be the quotient map from $\mathcal{G}B$ to $B$. Then, there exists a
lower semicontinuous convex function $\psi:\mathbb{R}^{p}\to(-\infty,+\infty]$
such that

$R(\mathbf{x})=q(\nabla\psi(\mathbf{x}))\quad(\mu\mbox{-a.e.}\leavevmode\nobreak\
\mathbf{x}).$

  2. (ii)

Let $S\sim{\rm Uniform}(\mathcal{G})$ be independent of $\mathbf{X}\sim\mu$
and $\mathbf{H}\sim\nu$. Let $\mu_{S}$ and $\nu_{S}$ denote the distributions
of $S\mathbf{X}$ and $S\mathbf{H}$, respectively. Then $\nabla\psi(\cdot)$ is
the $\mu_{S}$-a.e. unique gradient of convex function that pushes $\mu_{S}$ to
$\nu_{S}$.

  3. (iii)

$\nabla\psi$ is equivariant under the group action of $\mathcal{G}$, i.e.,
$\nabla\psi(Q\mathbf{x})=Q\nabla\psi(\mathbf{x})$ for $Q\in\mathcal{G}$ and
$\mathbf{x}\in\mathbb{R}^{p}$ being a differentiable point of $\psi$.

  4. (iv)

$\nabla\psi(\mathbf{X})\overset{a.s.}{=}S(\mathbf{X},R(\mathbf{X}))R(\mathbf{X})$
with
$S(\mathbf{x},\mathbf{h}):=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}$.
Hence, $\nabla\psi(\cdot)$ can be viewed as the population signed-rank map.

  5. (v)

If $\mu$ and $\nu$ have finite second moments, then $({\rm identity},R)\\#\mu$
is the unique solution to the population OT problem (2.5).

Let us discuss the conclusions in $(i)$-$(v)$ of the above result: $(i)$ shows
that the population rank map is essentially the gradient of a convex function
$\psi(\cdot)$ (cf. McCann’s geometric characterization of OT maps [73] under
the usual squared loss), up to a composition with the quotient map $q(\cdot)$
which brings it to $B$. The convex function $\psi(\cdot)$ is characterized in
$(ii)$ — its gradient pushes the symmetrized data distribution $\mu_{S}$ to
the symmetrized reference measure $\nu_{S}$, and it is unique (which follows
from the celebrated result in McCann, [73]). Although the result in [73] would
imply that $\nabla\psi(\cdot)$ is $\mu_{S}$-a.e. uniquely defined, Lemma B.3
in the Appendix shows that $\mu$ is absolutely continuous w.r.t. $\mu_{S}$,
and thus $\nabla\psi(\cdot)$ is also $\mu$-a.e. uniquely defined. In $(iii)$
and $(iv)$ of Theorem 2.1 we illustrate some of the ‘nice’ properties of
$\nabla\psi(\cdot)$: it respects the symmetry of $\mathcal{G}$, and turns out
to be the population signed-rank map, which further motivates the study of the
generalized (sample) signed-ranks $S_{n}(\cdot)R_{n}(\cdot)$. Note that
$(i)$-$(iv)$ in the above result do not assume any moment conditions on $\mu$
and $\nu$, and provides a geometric characterization of the rank map using
$c$-cyclical monotonicity. When second moments of $\mu$ and $\nu$ do exist,
$(v)$ shows that the population rank map is indeed the OT map in the sense of
minimizing (2.5).

######  Remark 2.6 (On the proof of Theorem 2.1).

The proof of the above result proceeds via checking the uniqueness of the
$c$-subdifferential set [101, Theorem 5.30] to show that the $c$-cyclically
monotone measure in $\Pi(\mu,\nu)$ is unique, and is given by a Monge map. It
crucially uses the structure of the cost function $c(\cdot,\cdot)$, e.g., a
$c$-convex function (defined in Section B.4) is still convex in the usual
sense, with its (almost everywhere) gradient being $\mathcal{G}$-equivariant.

Note that the minimizer
$S(\mathbf{x},\mathbf{h})=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}$
may not be unique in general. For example, when $\mathcal{G}={\rm O}(p)$,
there are many orthogonal matrices that map a given vector to a given
direction (see e.g., Figure 2(c)). However, in the proof of Theorem 2.1 we
show that the signed-rank $S(\mathbf{X},R(\mathbf{X}))R(\mathbf{X})$ is
$\mu$-a.s. unique — it is the point in the orbit of $R(\mathbf{X})$, i.e.,
$\\{QR(\mathbf{X}):Q\in\mathcal{G}\\}$, that is closest to $\mathbf{X}$.

With the population rank map $R(\cdot)$ and the signed-rank map
$\nabla\psi(\cdot)$ defined in Theorem 2.1, we state below a convergence
theorem for the sample generalized signs, ranks and signed-ranks. We assume
the following condition:

######  Assumption 2 (On weak convergence of $\nu_{n}$).

The empirical measure
$\nu_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{h}_{i}}$ converges weakly
to the reference distribution $\nu$.

The result below states that these sample quantities converge to their
population counterparts in the averaged $\rho$-loss. Here, we let
$\rho(\cdot,\cdot)$ be any continuous nonnegative function on
$\mathbb{R}^{p}\times\mathbb{R}^{p}$ satisfying: (i)
$\rho(\mathbf{x},\mathbf{x})=0$, $\forall\;\mathbf{x}\,\in\,\mathbb{R}^{p}$;
(ii) there exists $C_{1}>0$ such that $\rho(\mathbf{x},\mathbf{y})\leq
C_{1}\left(\rho(\mathbf{x},\mathbf{0})+\rho(\mathbf{y},\mathbf{0})\right)$,
$\forall\;\mathbf{x}$, $\mathbf{y}\in\mathbb{R}^{p}$; (iii) there exists
$C_{2}>0$ such that $\rho(Q\mathbf{x},\mathbf{0})\leq
C_{2}\left(\rho(\mathbf{x},\mathbf{0})+1\right)$,
$\forall\;\mathbf{x}\,\in\,\mathbb{R}^{p}$, $Q\in\mathcal{G}$. Examples of
$\rho(\cdot,\cdot)$ include the usual Euclidean distance $\|\cdot-\cdot\|$,
any norm on $\mathbb{R}^{p}$, and
$\rho(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|^{k}$ for any $k>0$.

######  Theorem 2.2 (Convergence to population generalized ranks, signs and
signed-ranks).

Suppose Assumptions 1 and 2 hold. We have:

  1. 1.

(Convergence of ranks) If the reference distribution $\nu$, which concentrates
on $B$, is chosen such that the quotient map $q:\mathcal{G}B\to B$ defined as
$q(Q\mathbf{x})=\mathbf{x}$ (for $\mathbf{x}\in B$, $Q\in\mathcal{G}$) is
continuous w.r.t. the usual Euclidean topology, then
$R(\cdot)=q(\nabla\psi(\cdot))$ is $\mu$-a.e. continuous. If furthermore,
$\mathbb{E}\rho(\mathbf{H}_{n},\mathbf{0})\to\mathbb{E}\rho(\mathbf{H},\mathbf{0})<\infty$
as $n\to\infty$, where $\mathbf{H}_{n}\sim\nu_{n}$ and $\mathbf{H}\sim\nu$,
then

$\frac{1}{n}\sum_{i=1}^{n}\rho(R_{n}(\mathbf{X}_{i}),R(\mathbf{X}_{i}))\overset{a.s.}{\longrightarrow}0,\quad{\rm
as}\ n\to\infty.$

  2. 2.

(Convergence of signs) If $\mathcal{G}$ acts freely on $\mathcal{G}B$, then
$S(\mathbf{x},R(\mathbf{x})):=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-R(\mathbf{x})\|^{2}$
is uniquely defined for $\mu$-a.e. $\mathbf{x}$, and for any continuous
function $\widetilde{\rho}(\cdot,\cdot)$ on
$\mathbb{R}^{p^{2}}\times\mathbb{R}^{p^{2}}$ such that
$\widetilde{\rho}(Q,Q)=0$, for all $Q\,\in\mathbb{R}^{p^{2}}$, we have

$\frac{1}{n}\sum_{i=1}^{n}\widetilde{\rho}(S_{n}(\mathbf{X}_{i}),S(\mathbf{X}_{i},R(\mathbf{X}_{i})))\overset{a.s.}{\longrightarrow}0,\quad{\rm
as}\ n\to\infty.$

  3. 3.

(Convergence of signed-ranks) If
$\mathbb{E}\rho(\mathbf{H}_{n},\mathbf{0})\to\mathbb{E}\rho(\mathbf{H},\mathbf{0})<\infty$
where $\mathbf{H}_{n}\sim\nu_{n}$, $\mathbf{H}\sim\nu$, then

$\frac{1}{n}\sum_{i=1}^{n}\rho(S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i}),\nabla\psi(\mathbf{X}_{i}))\overset{a.s.}{\longrightarrow}0,\quad{\rm
as}\ n\to\infty.$

A proof of the above result is given in Section B.4. To establish the
convergence of ranks, the continuity of $q$ is needed, while to establish the
convergence of $S_{n}(\mathbf{X}_{i})$, we need to restrict ourselves to
$\mathcal{G}$ with a free group action (which includes central symmetry and
sign symmetry) to make sure that $S_{n}(\mathbf{X}_{i})$ and
$S(\mathbf{X}_{i},R(\mathbf{X}_{i}))$ are a.s. uniquely defined. The
convergence of the signed-ranks requires the weakest assumption, again
motivating that test statistics based on the signed-ranks may be preferable.
This intuition will be confirmed in Section 3. Note that the above result
allows for the use of any ‘reasonable’ loss function while comparing the
sample and population signs, ranks and signed-ranks.

##  3 Distribution-Free Tests for Symmetry

In this section, we will define multivariate generalizations of the sign test
and the Wilcoxon signed-rank test that share analogous properties with their
classical counterparts (Sections 3.1-3.2). We then study the consistency of
our proposed tests (Section 3.3) and state results on their relative
efficiency w.r.t. Hotelling’s $T^{2}$ test (Section 3.4). We describe the
locally asymptotically optimal property of our generalized Wilcoxon signed-
rank (GWSR) test in Section 3.4.

###  3.1 Generalized Sign Test

Recall the hypothesis of $\mathcal{G}$-symmetry in (1.1), and that under ${\rm
H}_{0}$, the generalized signs
$S_{n}(\mathbf{X}_{1}),\ldots,S_{n}(\mathbf{X}_{n})$ are i.i.d. from ${\rm
Uniform}(\mathcal{G})$. Hence, any test of uniformity over $\mathcal{G}$ can
be applied with the generalized signs to define a multivariate analogue of the
sign test.

######  Example 3.1 (Chi-squared test).

If $\mathcal{G}=\\{g_{1},\ldots,g_{k}\\}$ is a finite group of size $k$, then
we can count the observed frequencies of each element in $\mathcal{G}$, i.e.,
$Y_{i}:=\sum_{j=1}^{n}I(S_{n}(\mathbf{X}_{j})=g_{i})$, $i=1,\ldots,k$. Under
${\rm H}_{0}$, $(Y_{1},\ldots,Y_{k})$ follows ${\rm
Multinomial}\left(n,\frac{1}{k}\mathbf{1}_{k}\right)$. Further, we can perform
an asymptotic Pearson’s chi-squared test:

$\widetilde{T}_{n}:=\sum_{i=1}^{k}\frac{(Y_{i}-n/k)^{2}}{n/k}\overset{d}{\longrightarrow}\chi^{2}_{k-1},\quad{\rm
as\ }n\to\infty.$

${\rm H}_{0}$ can be rejected for a large value of $\widetilde{T}_{n}$.

However, for a group with large size $k$ (e.g., $\mathcal{G}$ corresponding to
sign symmetry has size $2^{p}$, which can be very large for even moderate
$p$), the convergence of the chi-squared test statistic $\widetilde{T}_{n}$ to
its limiting distribution can be slow. Hence, in this paper we propose the
following generalized sign test:

$T_{n}:=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i}).$ (3.1)

We reject the null hypothesis ${\rm H}_{0}$ for large values of
$\|T_{n}\|^{2}_{F}$ where $\|\cdot\|_{F}$ denotes the matrix Frobenius norm.
The generalized sign test is distribution-free under ${\rm H}_{0}$ as the
signs $S_{n}(\mathbf{X}_{i})$ are i.i.d. from ${\rm Uniform}(\mathcal{G})$. We
give below a simple result on the asymptotic behavior of $\|T_{n}\|^{2}_{F}$
under ${\rm H}_{0}$. Note that this test statistic is also a direct
generalization of the classical sign test.

######  Proposition 3.1 (Generalized sign test).

We have the following consequences:

  1. 1.

Central symmetry: $S_{n}(\mathbf{X}_{i})$ are i.i.d. uniform over
$\mathcal{G}=\\{I,-I\\}$ under ${\rm H}_{0}$, and $T_{n}$ (in (3.1)) is thus
equivalent to a binomial random variable (up to a linear transformation) in
finite samples. The asymptotic distribution of $T_{n}$ follows from the usual
one-dimensional CLT:

$\frac{1}{p}\|T_{n}\|^{2}_{F}\overset{d}{\longrightarrow}\chi^{2}_{1},\quad{\rm
as}\ n\to\infty.$

  2. 2.

Sign symmetry: $S_{n}(\mathbf{X}_{i})$ are i.i.d. uniform over diagonal
orthogonal matrices under the null. Equivalently, $S_{n}(\mathbf{X}_{i})$ can
be viewed as being i.i.d. uniform over $\\{1,-1\\}^{p}$. In such a case,

$\|T_{n}\|^{2}_{F}\overset{d}{\longrightarrow}\chi^{2}_{p},\quad{\rm as}\
n\to\infty.$

  3. 3.

Spherical symmetry: $S_{n}(\mathbf{X}_{i})$ are i.i.d. uniform over the
orthogonal group ${\rm O}(p)$ under ${\rm H}_{0}$. In this case, our
generalized sign test behaves as:

${p}\|T_{n}\|^{2}_{F}\overset{d}{\longrightarrow}\chi^{2}_{p^{2}},\quad{\rm
as}\ n\to\infty.$

Possibly the simplest test of uniformity over ${\rm O}(p)$ is given by the
Rayleigh test [72, Section 13.2.2], which rejects the uniformity assumption
when $pn{\rm Tr}\left(\overline{S}^{\top}\overline{S}\right)$ is large, where
$\overline{S}:=\frac{1}{n}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})$. Note that this
is exactly the generalized sign test given in 3.1-3. We also observe that all
the above three cases reduce to the classical two-sided sign test when $p=1$.

###  3.2 Generalized Wilcoxon Signed-Rank (GWSR) Test

The GWSR statistic can naturally be defined as
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})$.
We also consider a more general score transformed version: for a score
function $J:B\to\mathbb{R}^{p}$ define

$\mathbf{W}_{n}:=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})J(R_{n}(\mathbf{X}_{i})).$
(3.2)

Note that $\mathbf{W}_{n}$ is distribution-free under $\textrm{H}_{0}$ (see
(1.1)) following 2.2. Apart from using its exact distribution in finite
samples, one can also use its asymptotic distribution as $n\to\infty$, which
yields a computationally simple test.

######  Definition 3.1 (Effective reference distribution).

We define the effective reference distribution (ERD), obtained with score
function $J(\cdot)$ and reference distribution $\nu$, as the distribution of
$SJ(\mathbf{H})$, where $S\sim{\rm Uniform}(\mathcal{G})$ is independent of
$\mathbf{H}\sim\nu$.

The ERD is $\mathcal{G}$-symmetric, i.e., $QSJ(\mathbf{H})$ has the same
distribution as $SJ(\mathbf{H})$, for all $Q\in\mathcal{G}$. Let $\Sigma_{\rm
ERD}$ be the covariance matrix of the ERD. The GWSR test rejects the null
hypothesis ${\rm H}_{0}$ for large values of

$\mathbf{W}_{n}^{\top}\Sigma_{\rm ERD}^{-1}\mathbf{W}_{n}.$ (3.3)

In the next few subsections we study the behavior of the GWSR test under the
null and under (local) contiguous alternatives. For this, we need a subset of
the following assumptions, which we list below for the convenience of the
reader.

We say $J:\mathcal{G}B\to\mathbb{R}^{p}$ is equivariant under the group action
of $\mathcal{G}$ if $J(Q\mathbf{x})=QJ(\mathbf{x})$ for all $Q\in\mathcal{G}$
and $\mathbf{x}\in\mathcal{G}B$. If $J:B\to\mathbb{R}^{p}$ can be extended to
an equivariant function on $\mathcal{G}B$, then $\mathbf{W}_{n}$ will be a
statistic based on the signed-ranks.

######  Assumption 3 (On the score function $J$).

$J:\mathcal{G}B\to\mathbb{R}^{p}$ is $\nu_{S}$-a.e. continuous and equivariant
under the group action of $\mathcal{G}$.

######  Assumption 4 (Moment convergence of $\nu_{n}$ to $\nu$).

Let
$\mathbf{H}_{n}\sim\nu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{h}_{i}}$
and $\mathbf{H}\sim\nu$.

  1. (i)

(1st moment)
$\mathbb{E}\|J(\mathbf{H}_{n})\|\to\mathbb{E}\|J(\mathbf{H})\|<\infty$ as
$n\to\infty$.

  2. (ii)

(2nd moment)
$\mathbb{E}\|J(\mathbf{H}_{n})\|^{2}\to\mathbb{E}\|J(\mathbf{H})\|^{2}<\infty$
as $n\to\infty$.

  3. (iii)

(2nd moments)
$\mathbb{E}|J(\mathbf{H}_{n})_{i}J(\mathbf{H}_{n})_{j}|\to\mathbb{E}|J(\mathbf{H})_{i}J(\mathbf{H})_{j}|<\infty$
as $n\to\infty$, for $i,j\in\\{1,\ldots,p\\}$.

######  Assumption 5 (On the group $\mathcal{G}$).

  1. (i)

$\mathbb{E}[{\rm Uniform}(\mathcal{G})]=\mathbf{0}_{p\times p}$; (ii)
$-I_{p}\in\mathcal{G}$.

Note that, under the condition that $\nu_{n}$ converges weakly to $\nu$ (see
2), Assumption 4-$(iii)$ implies Assumption 4-$(ii)$, and Assumption 4-$(ii)$
implies Assumption 4-$(i)$. By the strong law of large numbers, if $J\\#\nu$
has finite second moments, and $\nu_{n}$ is obtained from i.i.d. sampling from
$\nu$, then $\nu_{n}$ satisfies Assumption 4-$(iii)$. If $J(\cdot)$ is bounded
and continuous, then Assumption 4-$(iii)$ also holds by the dominated
convergence theorem, as long as $\nu_{n}$ converges weakly to $\nu$.
Similarly, Assumption 5-$(ii)$ implies Assumption 5-$(i)$, and central
symmetry, sign symmetry, spherical symmetry all satisfy Assumption 5-$(ii)$.

The following result (proved in Section B.3) shows that $\mathbf{W}_{n}$ has
an asymptotic normal distribution, mirroring the classical Wilcoxon signed-
rank statistic.

######  Theorem 3.1 (Asymptotic normality of $\mathbf{W}_{n}$).

Suppose Assumptions 2, 4-(iii) and 5-(i) hold, and that the score function
$J(\cdot)$ used in defining $\mathbf{W}_{n}$ (see (3.2)) is $\nu$-a.e.
continuous. Then, under ${\rm H}_{0}$,

$\mathbf{W}_{n}\overset{d}{\longrightarrow}N\left(\mathbf{0}_{p},\Sigma_{\rm
ERD}\right).$

###  3.3 Consistency of the Tests

If $\mathbf{X}$ is $\mathcal{G}$-symmetric, then by the equivariance of $\psi$
(see Theorem 2.1),
$Q\nabla\psi(\mathbf{X}){=}\nabla\psi(Q\mathbf{X})\overset{d}{=}\nabla\psi(\mathbf{X})$
for all $Q\in\mathcal{G}$. Hence $\nabla\psi(\mathbf{X})$ is also
$\mathcal{G}$-symmetric, and
$\mathbb{E}[J(\nabla\psi(\mathbf{X}))]=\mathbf{0}$ if 5-$(i)$ holds and
$J(\cdot)$ is equivariant.444For $S\sim{\rm Uniform}(\mathcal{G})$ independent
of $\mathbf{H}\sim\nu$, we have
$\mathbb{E}[J(\nabla\psi(\mathbf{X}))]=\mathbb{E}[J(S\mathbf{H})]=\mathbb{E}[SJ(\mathbf{H})]=\mathbb{E}[S]\mathbb{E}[J(\mathbf{H})]=\mathbf{0}$.
On the other hand, if $\mathbf{X}$ is not $\mathcal{G}$-symmetric, then
$\mathbb{E}[J(\nabla\psi(\mathbf{X}))]$ may not be $\mathbf{0}$. The following
result (Theorem 3.2; see Section B.4 for a proof) states that under this
assumption our GWSR test will be consistent.

######  Theorem 3.2 (Consistency of GWSR test).

Suppose Assumptions 1, 2, 3 and 4-(i) hold. Then the distribution-free GWSR
test (see (3.3)) is consistent against all alternatives for which
$\mathbb{E}[J(\nabla\psi(\mathbf{X}))]\neq\mathbf{0}$.

It is natural to ask if the condition
$\mathbb{E}[J(\nabla\psi(\mathbf{X}))]\neq\mathbf{0}$ holds for location shift
alternatives. The following result (see Section B.6 for a proof) shows that it
is indeed the case, thereby showing that the GWSR test (3.3) for
$\mathcal{G}$-symmetry (if $-I\in\mathcal{G}$) is consistent against location
shift alternatives of a centrally symmetric distribution.

######  Corollary 3.1 (Consistency in location shift model).

Suppose Assumptions 1, 2, 4-(i) and 5-(ii) hold. Suppose
$J(\mathbf{x})=\mathbf{x}$, and $\mathbf{X}$ is a centrally symmetric
distribution shifted by $\mathbf{\Delta}\neq\mathbf{0}$, i.e.,

$\mathbf{X}-\mathbf{\Delta}\stackrel{{\scriptstyle
d}}{{=}}\mathbf{\Delta}-\mathbf{X}.$

If $\psi$ (defined in Theorem 2.1) is strictly convex555$\psi$ is said to be
strictly convex on $U$ if for any $\mathbf{x}$, $\mathbf{y}\in U$ and
$\alpha\in(0,1)$,
$\psi(\alpha\mathbf{x}+(1-\alpha)y)<\alpha\psi(\mathbf{x})+(1-\alpha)\psi(\mathbf{y})$.
on an open set of positive $\mu$-measure, then the GWSR test is consistent.

Let us now discuss the consistency properties of the generalized sign test.
Recall that $S_{n}(\mathbf{X}_{i})\equiv
S(\mathbf{X}_{i},R_{n}(\mathbf{X}_{i}))$ where
$S(\mathbf{x},\mathbf{h})=\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}$.
When we have a free group action, $S(\mathbf{x},\mathbf{h})$ will be $({\rm
identity},R)\\#\mu$-a.e. unique (see Theorem 2.2). For example:

  1. 1.

(Central symmetry) $S(\mathbf{x},\mathbf{h})={\rm
sign}(\mathbf{x}^{\top}\mathbf{h})I_{p}$ is unique provided that
$\mathbf{x}^{\top}\mathbf{h}\neq 0$.

  2. 2.

(Sign symmetry) $S(\mathbf{x},\mathbf{h})={\rm
diag}\left(\textbf{sign}(\mathbf{x}\odot\mathbf{h})\right)$ is unique provided
$\mathbf{x}\odot\mathbf{h}\neq\mathbf{0}$, where for
$\mathbf{a},\mathbf{b}\in\mathbb{R}^{p}$, $\mathbf{a}\odot\mathbf{b}$ denotes
$(a_{1}b_{1},\ldots,a_{p}b_{p})$, and $\textbf{sign}(\cdot)$ is the component
sign.

######  Remark 3.1 (Spherical symmetry).

For spherical symmetry, the minimizer $S(\mathbf{x},\mathbf{h})$ is not unique
and needs to be chosen uniformly at random from the set of all minimizers. We
can select the minimizer in the following way, by introducing an independent
Gaussian vector $\varepsilon\sim N(\mathbf{0},I_{p(p-1)})$: Suppose
$\mathbf{h},\mathbf{x}\neq\mathbf{0}$. Let
$\mathbf{v}=\frac{\mathbf{h}}{\|\mathbf{h}\|}$,
$\mathbf{w}=\frac{\mathbf{x}}{\|\mathbf{x}\|}$, and $V$ and $W$ be
$p\times(p-1)$ matrices such that $V^{\top}V=W^{\top}W=I_{p-1}$,
$V^{\top}\mathbf{v}=W^{\top}\mathbf{w}=\mathbf{0}$. Generate
$S(\mathbf{x},\mathbf{h})\equiv S(\mathbf{x},\mathbf{h},\varepsilon)$ as a
function of $\mathbf{x},\mathbf{h},\varepsilon$:

  1. (a)

Set $\mathbf{v}^{\top}$ as the first row and extend it to a $p\times p$
orthogonal matrix $[\mathbf{v}\ V]^{\top}\in{\rm O}(p)$.

  2. (b)

Set $\mathbf{w}$ as the first column and add $p(p-1)$ i.i.d. standard normal
random variables $\varepsilon\in\mathbb{R}^{p(p-1)}$ to form a $p\times p$
matrix $[\mathbf{w}\ \varepsilon]$. Apply Gram-Schmidt orthogonalization to
the columns of $[\mathbf{w}\ \varepsilon]$ (in the order of the columns) to
obtain ${\rm GS}([\mathbf{w}\ \varepsilon])\in{\rm O}(p)$.

  3. (c)

Set $S(\mathbf{x},\mathbf{h},\varepsilon)={\rm GS}([\mathbf{w}\
\varepsilon])[\mathbf{v}\ V]^{\top}$.

The validity of the above procedure is shown in Section A.2.

The following result (proved in Section B.4) gives sufficient conditions for
the consistency of the generalized sign test for the three prime examples of
$\mathcal{G}$ considered in this paper, corresponding to central, sign and
spherical symmetry.

######  Theorem 3.3 (Consistency of the generalized sign test).

Suppose Assumptions 1 and 2 hold. Then, the generalized sign test, which
rejects $\mathrm{H}_{0}$ for large values of $\|T_{n}\|_{F}$, is consistent if

  * •

(Central/Sign symmetry) $\mathcal{G}$ acts freely on $\mathcal{G}B$, and
$\mathbb{E}[S(\mathbf{X},R(\mathbf{X}))]\neq\mathbf{0}$.

  * •

(Spherical symmetry) $\mathcal{G}={\rm O}(p)$, $\nu(\\{\mathbf{0}\\})=0$, and
$\mathbb{E}[S(\mathbf{X},R(\mathbf{X}),\varepsilon)]\neq\mathbf{0}$, where
$S(\mathbf{x},\mathbf{h},\varepsilon)$ is given in the above construction
(which requires $\mathbf{h}\neq\mathbf{0}$).

Note that even in the univariate setting, the classical sign test may not be
consistent against location shift alternatives if the symmetric distribution
has no mass in a neighborhood of the origin.666For example, if $X\sim{\rm
Uniform}\left([-2,-1]\cup[1,2]\right)$, then the sign test will be powerless
against a location shift by $\delta<1$. In fact, the condition
$\mathbb{E}[S(\mathbf{X},R(\mathbf{X}))]\neq\mathbf{0}$ for consistency
mentioned in Theorem 3.3 directly generalizes its one-dimensional counterpart;
see remark below.

######  Remark 3.2 (Consistency of the proposed tests when $p=1$).

If $p=1$ and $\nu$ is concentrated on $(0,\infty)$, then by Theorem 3.3 the
generalized sign test is consistent against alternatives where
$\mathbb{E}[{\rm sign}(X)]\neq 0$. This coincides with the condition needed
for the consistency of the classical two-sided sign test [see e.g., 65,
Section 2.3]. If $\nu={\rm Uniform}(0,1)$, then $R(x)=F_{|X|}(|x|)$, where
$F_{|X|}(\cdot)$ is the cumulative distribution function of $|X|$, and
$\nabla\psi(x)={\rm sign}(x)F_{|X|}(|x|)$. Theorem 3.2 implies GWSR test with
identity score function is consistent against $\mathbb{E}[{\rm
sign}(X)F_{|X|}(|X|)]\neq 0$, which is equivalent to the condition for the
consistency of the classical Wilcoxon signed-rank test, i.e.,
$\mathbb{P}(X_{1}+X_{2}>0)\neq\frac{1}{2}$, where $X_{1},X_{2}$ are i.i.d.
from $\mu$; see Section B.5 for a proof.

###  3.4 Efficiency of the GWSR Test

In this subsection we study the asymptotic relative efficiency (ARE) of our
GWSR test w.r.t. the Hotelling’s $T^{2}$ test. We first introduce the
following lemma (see Section B.7 for a proof), which states that under ${\rm
H}_{0}$, we can replace the empirical signed-ranks by their population
counterparts, while keeping the limiting distribution of the GWSR statistic
unchanged. Results of this type are sometimes referred to as Hájek
representation (cf. Hájek et al., [42, Theorem 1, Section 6.1.7]) and have
appeared in a number of recent works on applications of multivariate ranks
defined via OT [96, 44].

######  Lemma 3.1 (Hájek representation).

Suppose Assumptions 1, 2, 3, 4-(ii) and 5-(i) hold. Then, under
$\textrm{H}_{0}$,

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})J(R_{n}(\mathbf{X}_{i}))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S(\mathbf{X}_{i},R(\mathbf{X}_{i}))J(R(\mathbf{X}_{i}))\overset{L^{2}}{\longrightarrow}0.$

The above lemma together with Le Cam’s third lemma implies the following CLT
under contiguous alternatives; see Section B.8 for a proof.

######  Theorem 3.4 (Asymptotic distribution under local alternatives).

Let $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ be i.i.d. with Lebesgue density
$f(\mathbf{\cdot}-{\bm{\theta}})$ on $\mathbb{R}^{p}$, where $f$ is the
density of a $\mathcal{G}$-symmetric distribution and
$\bm{\theta}\in\mathbb{R}^{p}$. Consider testing:

${\rm H}_{0}:{\bm{\theta}}=\mathbf{0}_{p}\qquad\textrm{versus}\qquad{\rm
H}_{1}:{\bm{\theta}}=\frac{{\bm{\xi}}_{n}}{\sqrt{n}},$ (3.4)

where ${\bm{\xi}}_{n}\to{\bm{\xi}}\in\mathbb{R}^{p}$ as $n\to\infty$. Suppose
Assumptions 1, 2, 3, 4-(ii) and 5-(i) hold. Under the assumption of
differentiability under quadratic mean of the parametric family
$\\{f(\cdot-{\bm{\theta}})\\}_{{\bm{\theta}}\in\mathbb{R}^{p}}$ (see Section
A.1 for the details), we have under ${\rm
H}_{1}:{\bm{\theta}}=\frac{{\bm{\xi}}_{n}}{\sqrt{n}}$,

$\mathbf{W}_{n}\overset{d}{\longrightarrow}N\left({\bm{\gamma}},\Sigma_{\rm
ERD}\right),\quad\qquad\mbox{where}\;\;{\bm{\gamma}}:=-\mathbb{E}_{{\rm
H}_{0}}\left[J(\nabla\psi(\mathbf{X}))\frac{{\bm{\xi}}^{\top}\nabla
f(\mathbf{X})}{f(\mathbf{X})}\right]\in\mathbb{R}^{p}.$

The above theorem implies that

$\mathbf{W}_{n}^{\top}\Sigma_{\rm
ERD}^{-1}\mathbf{W}_{n}\overset{d}{\longrightarrow}\left\|\Sigma_{\rm
ERD}^{-1/2}{\bm{\gamma}}+N(\mathbf{0},I_{p})\right\|^{2},$

which is a non-central $\chi^{2}$-distribution with $p$ degrees of freedom and
non-centrality parameter $\|\Sigma_{\rm ERD}^{-1/2}{\bm{\gamma}}\|^{2}$. Note
that this limiting distribution does not depend on $\mathcal{G}$ directly; it
only depends on $J(\cdot)$, $\nabla\psi(\cdot)$ (which further depends on
$\nu_{S}$; see Theorem 2.1-$(ii)$), ${\bm{\xi}}\in\mathbb{R}^{p}$, and the
distribution of $\mathbf{X}$ under ${\rm H}_{0}$. We compare this test with
the Hotelling’s $T^{2}$ statistic given by

$T^{2}=n\overline{\mathbf{X}}_{n}^{\top}S_{n}^{-1}\overline{\mathbf{X}}_{n},$
(3.5)

where
$S_{n}:=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{X}_{i}-\overline{\mathbf{X}}_{n})(\mathbf{X}_{i}-\overline{\mathbf{X}}_{n})^{\top}$
is the sample covariance matrix. Observe that as $n\to\infty$,
$S_{n}\overset{p}{\longrightarrow}\Sigma_{\mathbf{X}}:=\mathbb{E}[\left(\mathbf{X}-\mathbb{E}\mathbf{X}\right)\left(\mathbf{X}-\mathbb{E}\mathbf{X}\right)^{\top}]$,
if the latter is finite. The absolute continuity of $\mathbf{X}\sim\mu$
implies that $\Sigma_{\mathbf{X}}$ is positive definite. Under the sequence of
contiguous alternatives ${\bm{\theta}}_{n}\equiv{\bm{\xi}}_{n}/\sqrt{n}$, we
have

$\displaystyle\sqrt{n}S_{n}^{-1/2}\overline{\mathbf{X}}_{n}-\Sigma_{\mathbf{X}}^{-1/2}{\bm{\xi}}=\sqrt{n}S_{n}^{-1/2}\left(\overline{\mathbf{X}}_{n}-{\bm{\xi}}_{n}/\sqrt{n}\right)+(S_{n}^{-1/2}{\bm{\xi}}_{n}-\Sigma_{\mathbf{X}}^{-1/2}{\bm{\xi}})\overset{d}{\longrightarrow}N(\mathbf{0},I_{p}).$

Hence, under this contiguous alternative, the Hotelling’s $T^{2}$ statistic
has non-central $\chi^{2}$-distribution with $p$ degrees of freedom and non-
centrality parameter $\|\Sigma_{\mathbf{X}}^{-1/2}{\bm{\xi}}\|^{2}$.

######  Definition 3.2 (Asymptotic relative efficiency).

For ${\bm{\xi}}\in\mathbb{R}^{p}\backslash\\{\mathbf{0}\\}$, consider testing

${\rm H}_{0}:{\bm{\theta}}=\mathbf{0}_{p}\qquad\textrm{versus}\qquad{\rm
H}_{1}:{\bm{\theta}}=\varepsilon_{n}{\bm{\xi}}_{n},\quad{\rm where\
}\varepsilon_{n}\to 0,\ {\bm{\xi}}_{n}\to{\bm{\xi}}$

in the location shift model $f(\mathbf{\cdot}-{\bm{\theta}})$, where $f$ is
the density of a $\mathcal{G}$-symmetric distribution on $\mathbb{R}^{p}$. For
$0<\alpha<\beta<1$, let $N_{1}(\alpha,\beta,{\bm{\xi}})$ be the minimum number
of samples needed for the (asymptotic) level $\alpha$ test based on
$\mathbf{W}_{n}$ to achieve power $\beta$ when ${\bm{\theta}}={\bm{\xi}}$,
i.e.,

$N_{1}\left(\alpha,\beta,{\bm{\xi}}\right):=\min\left\\{K\geq
1:\mathbb{P}_{{\bm{\theta}}={\bm{\xi}}}\left(\mathbf{W}_{K}^{\top}\Sigma_{\rm
ERD}^{-1}\mathbf{W}_{K}\geq\chi_{p}^{2}(\alpha)\right)\geq\beta\right\\},$

where $\chi_{p}^{2}(\alpha)\in\mathbb{R}$ satisfies
$\mathbb{P}\left(\chi_{p}^{2}\geq\chi_{p}^{2}(\alpha)\right)=\alpha$.
Similarly, let $N_{2}(\alpha,\beta,{\bm{\xi}})$ denote the minimum number of
samples needed for the (asymptotic) level $\alpha$ test based on the
Hotelling’s $T^{2}$ statistic to achieve power $\beta$ when
${\bm{\theta}}={\bm{\xi}}$. The asymptotic relative efficiency (ARE) of the
GWSR test w.r.t. Hotelling’s $T^{2}$, in the direction ${\bm{\xi}}$, is
defined as:

${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})=\lim_{n\to\infty}\frac{N_{2}\left(\alpha,\beta,\varepsilon_{n}{\bm{\xi}}_{n}\right)}{N_{1}\left(\alpha,\beta,\varepsilon_{n}{\bm{\xi}}_{n}\right)}.$

The ARE, in general can depend on $\alpha$, $\beta$, $\\{\varepsilon_{n}\\}$
and $\\{{\bm{\xi}}_{n}\\}$. However, if the tests have asymptotically non-
central chi-squared distributions with the same degrees of freedom, as in our
case here (such that Theorem 3.4 holds), then the ARE is free of
$\alpha,\beta,\\{\varepsilon_{n}\\}$, and is the squared ratio of the
corresponding non-centrality parameters (see Section B.9 for a proof):

${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})=\frac{\|\Sigma_{\rm
ERD}^{-1/2}{\bm{\gamma}}\|^{2}}{\|\Sigma_{\mathbf{X}}^{-1/2}{\bm{\xi}}\|^{2}}.$

The above ARE turns out to be lower-bounded for some sub-families of
multivariate distributions.

In the one-dimensional case, when $f(\cdot)$ is the density of
$N(0,\sigma^{2})$ for some unknown $\sigma>0$, the relative efficiency of the
Wilcoxon signed-rank test against the $t$-test is $3/\pi\approx 0.95$ [57].
The following proposition (see Section B.10 for a proof) generalizes this
result. A Chernoff-Savage-type result is also provided, which illustrates that
the GWSR test can be no worse than the Hotelling’s $T^{2}$ test, in terms of
ARE, if the Gaussian reference distribution (with identity score function) is
used.

######  Proposition 3.2 (Gaussian case).

Suppose Assumptions 1, 2, 4-(ii) and 5-(i) hold. Suppose $f(\cdot)$ is the
density of a multivariate normal distribution $N(\mathbf{0}_{p},\Sigma)$ for
some unknown positive definite covariance matrix $\Sigma$.

  1. 1.

If ${\rm ERD}=N(\mathbf{0},I)$, with $J(\mathbf{x})=\mathbf{x}$, then for all
${\bm{\xi}}\neq\mathbf{0}\in\mathbb{R}^{p}$, ${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})=1$.

  2. 2.

If ${\rm ERD}={\rm Uniform}(-1,1)^{p}$, obtained from
$J(\mathbf{x})=(2\Phi(x_{1})-1,\ldots,2\Phi(x_{p})-1)^{\top}$, where $\Phi$ is
the cumulative distribution function (c.d.f.) of the standard normal
distribution777Note that this ERD excludes testing for spherical symmetry as
$J(\cdot)$ is not equivariant under the group action of ${\rm O}(p)$., then
for all ${\bm{\xi}}\neq\mathbf{0}$, ${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})={3}/{\pi}\approx
0.95$.

  3. 3.

If ${\rm ERD}$ is the spherical uniform distribution888That is, the product of
the uniform distribution over the unit sphere with the uniform distribution
over the unit interval of distances to the origin [43, 95, 45, 44]., with
$J(\mathbf{x})=\mathbf{x}$, then for all ${\bm{\xi}}\neq\mathbf{0}$,

${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})=\kappa_{p}\geq\left\\{\begin{aligned}
&0.95,\quad\quad{\rm for\ }p<5,\\\ &0.648,\quad\ \ {\rm for\ all\
}p,\end{aligned}\right.$

(see Section B.10 for the exact value of $\kappa_{p}$). In particular,
$\kappa_{1}=3/\pi$ reduces to the classical ARE of the Wilcoxon’s signed-rank
test against the $t$-test [57].

Note that the above result is valid for all possible choices of $\mathcal{G}$
as long as the ERD is of the specified form. We will observe this phenomenon
for all results in this section. A remarkable result of the classical Wilcoxon
signed-rank test in one-dimension is the Hodges-Lehmann phenomenon, which
states that the relative efficiency of the classical Wilcoxon signed-rank test
w.r.t. the $t$-test for location shift alternatives never falls below 0.864
[57]. The following theorem (see Section B.11 for a proof) provides a
multivariate analogue to this result, albiet in a more restrictive setting —
where the components of the multivariate vector are assumed to be independent.

######  Theorem 3.5 (Independent components case).

Under the assumptions of Theorem 3.4, suppose further that
$f(\mathbf{x})=\prod_{i=1}^{p}\widetilde{f}_{i}(x_{i})$, for
$\mathbf{x}=(x_{1},\ldots,x_{p})\in\mathbb{R}^{p}$, where
$\widetilde{f}_{1},\ldots,\widetilde{f}_{p}$ are univariate Lebesgue densities
with finite non-zero variances.

  1. 1.

If the ERD is ${\rm Uniform}(-1,1)^{p}$, with $J(\mathbf{x})=\mathbf{x}$, then
${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})\geq\frac{108}{125}\approx
0.864$.

Further, equality in the ARE lower bound holds if and only if for all
$\xi_{i}\neq 0$, $\widetilde{f}_{i}$ has density:

$\widetilde{f}_{i}(x)=\frac{3}{20\sqrt{5}\sigma_{i}^{3}}(5\sigma_{i}^{2}-x^{2})I(|x|\leq\sqrt{5}\sigma_{i}),\quad{\rm
for\ some\ }\sigma_{i}>0.$

  2. 2.

If the ERD is $N(\mathbf{0},I)$, with $J(\mathbf{x})=\mathbf{x}$, then ${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})\geq 1$.

Further, equality holds if and only if for all $\xi_{i}\neq 0$,
$\widetilde{f}_{i}$ is the density of a normal distribution.

The ARE of the GWSR test w.r.t. the Hotelling’s $T^{2}$ test can also be
lower-bounded for another rich class of multivariate probability
distributions, namely the class of elliptically symmetric distributions; this
is stated in the theorem below (see Section B.12 for a proof).

######  Theorem 3.6 (Elliptically symmetric case).

Under the assumptions of Theorem 3.4, suppose further that $\mathbf{X}$ has
finite second moments under ${\rm H}_{0}$ and

$f(\mathbf{x})\propto({\rm
det}(\Sigma))^{-\frac{1}{2}}\underline{f}\left(\mathbf{x}^{\top}\Sigma^{-1}\mathbf{x}\right),\qquad\mbox{
for }\mathbf{x}\in\mathbb{R}^{p},$

for some unknown positive definite matrix $\Sigma$, and some unknown function
$\underline{f}:\mathbb{R}^{+}\to\mathbb{R}^{+}$. Under standard regularity
conditions on the smoothness and integrability of $\underline{f}$ [25, 43]
(see Section B.12 for details), we have

  1. 1.

If the ERD is the spherical uniform distribution with
$J(\mathbf{x})=\mathbf{x}$, then:

${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})\geq\frac{81}{500}\cdot\frac{(\sqrt{2p-1}+1)^{5}}{p^{2}(\sqrt{2p-1}+5)}\geq
0.648.$

See Section B.12 for the distribution that attains the ARE lower bound above.

  2. 2.

If the ERD is $N(\mathbf{0},I)$ with $J(\mathbf{x})=\mathbf{x}$, then ${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})\geq 1$.

Equality above is attained if and only if $f(\cdot)$ is a multivariate normal
distribution.

In the following, we consider a class of generative models used in blind
source separation [16, 23, 97, 25] and in independent component analysis [61,
90, 5] and derive similar lower bounds on the ARE (see Section B.13 for a
proof).

######  Theorem 3.7 (Generative model for blind source mixing).

Under the assumptions of Theorem 3.4, suppose further that
$\mathbf{X}=A\mathbf{W}$ under $\mathrm{H}_{0}$ for some unknown orthogonal
matrix $A$ and $\mathbf{W}$ having independent components with Lebesgue
density $\widetilde{f}(\mathbf{w})=\prod_{i=1}^{p}\widetilde{f}_{i}(w_{i})$,
where $\mathbf{w}=(w_{1},\ldots,w_{p})$ and
$\widetilde{f}_{1},\ldots,\widetilde{f}_{p}$ are univariate densities. Suppose
the ERD is $N(\mathbf{0},I)$, with $J(\mathbf{x})=\mathbf{x}$, and
$\mathbf{X}$ has finite second moments under ${\rm H}_{0}$. Then:

${\rm ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n};{\bm{\xi}})\geq 1.$

The equality is attained if and only if $\widetilde{f}_{i}$ is the density of
a normal distribution whenever $(A^{\top}{\bm{\xi}})_{i}\neq 0$,
$i=1,2,\ldots,p$.

Note that in all of the above ARE results, there is no restriction on
$\mathcal{G}$ as long as the ERD is of the required form. Theorem 3.7 again
illustrates the superiority of using a Gaussian ERD with the identity score
function. Based on this fact, we recommend some choices of the reference
measure $\nu$ below for different $\mathcal{G}$’s.

######  Remark 3.3 (Examples of the reference measure $\nu$).

Here we recommend some choices of the reference measure $\nu$, based on the
canonical choice of the score function $J(\mathbf{x})=\mathbf{x}$:

  1. 1.

Our first recommendation is a reference distribution $\nu$ with Gaussian ERD —
$N(\mathbf{0},I_{p})$. For example, for central symmetry, we can take $\nu$ to
be the distribution of $(|Z_{1}|,Z_{2},\ldots,Z_{p})$ (concentrated on
$B=(0,\infty)\times\mathbb{R}^{p-1}$) where $Z_{i}$’s are i.i.d. standard
normal; for sign symmetry, we can take $\nu$ as the distribution of
$(|Z_{1}|,|Z_{2}|,\ldots,|Z_{p}|)$ (here $B=(0,\infty)^{p}$). For spherical
symmetry, we can take $\nu$ to be the distribution of
$\left(\sqrt{Z_{1}^{2}+\ldots+Z_{p}^{2}},{0},\ldots,{0}\right)$ and
$B=(0,\infty)\times\\{0\\}^{p-1}$. Using Gaussian ERD is particularly powerful
as the ARE of the GWSR test w.r.t. the Hotelling’s $T^{2}$ test never falls
below 1 for a large class of models (see Theorems 3.5-3.7).

  2. 2.

Our second recommendation is a reference distribution $\nu$ with uniform ERD —
${\rm Uniform}(-1,1)^{p}$. For example, for central symmetry, we can take
$\nu$ to be the distribution of $(|U_{1}|,U_{2},\ldots,U_{p})$ (concentrated
on $B=(0,1)\times(-1,1)^{p-1}$) where $U_{i}$’s are i.i.d. ${\rm
Uniform}(-1,1)$; for sign symmetry, we can take $\nu$ as the distribution of
$(|U_{1}|,|U_{2}|,\ldots,|U_{p}|)$ (thus $B=(0,1)^{p}$). Note that ${\rm
Uniform}(-1,1)^{p}$ is not spherically symmetric, so this ERD is not
permissible for testing spherical symmetry.

  3. 3.

Our third recommendation is a reference distribution $\nu$ with spherical
uniform ERD, which reduces to ${\rm Uniform}(-1,1)$ when $p=1$, and has been
heavily advocated in the literature of distribution-free inference using OT
[43, 95, 45, 44]. Reference measures for central symmetry, sign symmetry, and
spherical symmetry can also be constructed similarly as described above for
the Gaussian ERD.

We have so far illustrated lower bounds on the ARE of the GWSR test against
the Hotelling’s $T^{2}$ test. A natural question arises whether the
generalized sign test would also share similar properties? A positive result
holds for sign symmetry, where the generalized signs coincide with the
component-wise signs, and was studied in Bickel, [12]. The following theorem
is a direct consequence of Bickel, [12, Equation (5.5)].

######  Theorem 3.8 (ARE for the generalized sign test under sign symmetry
[12]).

Suppose the reference distribution $\nu_{n}$ is concentrated on
$(0,+\infty)^{p}$ for all $n$. Assume a Lebesgue density $f(\cdot)$ is sign
symmetric with non-degenerate covariance matrix, and each marginal density
$f_{i}$ of $f$ has $0$ as its mode, and is continuous at $0$. Consider the
model $f(\cdot-{\bm{\theta}})$ and $H_{0}$: ${\bm{\theta}}=\mathbf{0}_{p}$
against contiguous alternatives. Then the relative efficiency of the
generalized sign test (using sign symmetry) with respect to Hotelling’s
$T^{2}$-test never falls below $1/3$.

Table 1: Relative efficiencies of the generalized sign test (for sign symmetry) versus the Hotelling’s $T^{2}$ test for some sign symmetric distributions. The minimally possible value of the relative efficiency is $1/3$. Distribution | ARE (sign/Hotelling’s $T^{2}$)
---|---
Laplace: $\prod_{i=1}^{p}{\rm Laplace}(0,b_{i})$ | 2
Normal: $\prod_{i=1}^{p}N(0,\sigma_{i}^{2})$ | $2/\pi$
Logistic: $\prod_{i=1}^{p}{\rm Logistic}(0,s_{i})$ | $\pi^{2}/12$
Uniform: $\prod_{i=1}^{p}{\rm Uniform}(-c_{i},c_{i})$ | 1/3 (lower bound)

In Table 1, the relative efficiencies of the generalized sign test (for sign
symmetry) versus the Hotelling’s $T^{2}$ test for some distributions are given
(using the formula [12, Equation (5.2)]), which is exactly the same as in one-
dimensional case [99, Table 14.1]. Besides, by designing a large density at
$0$, the relative efficiency can be arbitrarily large, which would strongly
favor the sign test over the Hotelling’s $T^{2}$ test. However, For other
types of symmetry, there is in general no lower bound of ARE. The ARE could
depend on $\alpha$, $\beta$, the reference distribution, the direction of
${\bm{\xi}}$, and the dimension $p$. From our simulations in Section 4, we
also find that the generalized sign test using sign symmetry tend to have a
better performance than using the central symmetry or the spherical symmetry.
Hence, we would recommend using sign symmetry as the primary choice when
performing the generalized sign test.

So far, we have shown that for natural (and practical) choices of sub-families
of location shift alternatives and suitable score functions (mostly
$J(\mathbf{x})=\mathbf{x}$), the GWSR test performs at least as well as the
Hotelling’s $T^{2}$ test. In the following result (proved in Section B.14), we
show, through two separate results, that there exists a score transformed
Wilcoxon’s signed-rank test (with appropriately chosen score function) that is
locally asymptotically optimal, i.e., it has maximum local power (against
contiguous alternatives) among all tests with fixed Type I error.

######  Theorem 3.9 (Local asymptotic optimality).

Let $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ be a sample from $P_{\bm{\theta}}$
with Lebesgue density $f(\cdot-{\bm{\theta}})$, where $f$ is
$\mathcal{G}$-symmetric about ${\bm{\theta}}_{0}:=\mathbf{0}_{p}$. Under the
assumptions in Theorem 3.4, let $I_{{\bm{\theta}}_{0}}$ be the nonsingular
Fisher information matrix at ${\bm{\theta}}_{0}=\mathbf{0}_{p}$. Suppose
$\nu_{S}$ has a Lebesgue density. Let
$\psi^{*}(\mathbf{y}):=\sup_{\mathbf{x}\in\mathbb{R}^{p}}\\{\mathbf{y}^{\top}\mathbf{x}-\psi(\mathbf{x})\\}$
be the Legendre-Fenchel transform of $\psi$, and suppose

$J(\mathbf{x}):=-\frac{\nabla
f(\nabla\psi^{*}(\mathbf{x}))}{f(\nabla\psi^{*}(\mathbf{x}))},\qquad\mbox{for}\;\;\mathbf{x}\in\mathcal{G}B,$
(3.6)

is used in defining $\mathbf{W}_{n}$ (via (3.2)).

  1. 1.

(Locally maximin optimal test) Let
${\bm{\xi}}\neq\mathbf{0}\in\mathbb{R}^{p}$. For any sequence of level
$\alpha\in(0,1)$ tests $\\{\phi_{n}\\}_{n\geq 1}$ for testing

${\rm H}_{0}:{\bm{\theta}}=\mathbf{0}_{p}\qquad{\rm versus}\qquad{\rm
H}_{1}:{\bm{\theta}}=\frac{{\bm{\xi}}}{\sqrt{n}}$ (3.7)

with power function
$\pi_{n}({\bm{\theta}}):=\mathbb{E}_{\bm{\theta}}[\phi_{n}]$, such that
$\pi_{n}\left(\frac{{\bm{\xi}}}{\sqrt{n}}\right)$ converges under ${\rm
H}_{1}$ for any ${\bm{\xi}}\in\mathbb{R}^{p}$, we have

$\displaystyle\inf_{{\bm{\xi}}:\|I^{1/2}_{{\bm{\theta}}_{0}}{\bm{\xi}}\|=c}\lim_{n\to\infty}\pi_{n}\left(\frac{{\bm{\xi}}}{\sqrt{n}}\right)\leq\inf_{{\bm{\xi}}:\|I^{1/2}_{{\bm{\theta}}_{0}}{\bm{\xi}}\|=c}\lim_{n\to\infty}\mathbb{P}_{{\bm{\theta}}_{0}+\frac{{\bm{\xi}}}{\sqrt{n}}}\left(\mathbf{W}_{n}^{\top}\Sigma_{\rm
ERD}^{-1}\mathbf{W}_{n}\geq\chi_{p}^{2}(\alpha)\right),$

for all $c>0$.

  2. 2.

(Locally most powerful test) Let $\zeta:\mathbb{R}^{p}\to\mathbb{R}$ be
differentiable at ${\bm{\theta}}_{0}=\mathbf{0}_{p}$ and
$\zeta({\bm{\theta}}_{0})=0$. Let ${\bm{\xi}}\in\mathbb{R}^{p}$ be such that
${\bm{\xi}}^{\top}\nabla\zeta({\bm{\theta}}_{0})>0$. For any sequence of level
$\alpha\in(0,1)$ tests $\\{\phi_{n}\\}_{n\geq 1}$ for

${\rm H}_{0}:\zeta({\bm{\theta}})\leq 0\qquad{\rm versus}\qquad{\rm
H}_{1}:\zeta({\bm{\theta}})>0$ (3.8)

with power function
$\pi_{n}({\bm{\theta}})=\mathbb{E}_{\bm{\theta}}[\phi_{n}]$, we have

$\displaystyle\limsup_{n\to\infty}\pi_{n}\left(\frac{{\bm{\xi}}}{\sqrt{n}}\right)\leq\lim_{n\to\infty}\mathbb{P}_{{\bm{\theta}}_{0}+\frac{{\bm{\xi}}}{\sqrt{n}}}\left(\frac{\nabla\zeta({\bm{\theta}}_{0})^{\top}I_{{\bm{\theta}}_{0}}^{-1}\mathbf{W}_{n}}{\sqrt{\nabla\zeta({\bm{\theta}}_{0})^{\top}I_{{{\bm{\theta}}_{0}}}^{-1}\nabla\zeta({\bm{\theta}}_{0})}}\geq
z_{\alpha}\right),$

where $z_{\alpha}$ satisfies $\Phi(z_{\alpha})=1-\alpha$, and
$\frac{\nabla\zeta({\bm{\theta}}_{0})^{\top}I_{{\bm{\theta}}_{0}}^{-1}\mathbf{W}_{n}}{\sqrt{\nabla\zeta({\bm{\theta}}_{0})^{\top}I_{{{\bm{\theta}}_{0}}}^{-1}\nabla\zeta({\bm{\theta}}_{0})}}\geq
z_{\alpha}$ is the appropriate (asymptotic) level $\alpha$ test based on the
GWSR statistic $\mathbf{W}_{n}$ for (3.8).

The first result above shows that the GWSR test maximizes the minimum (local)
power over the contour
$\\{{\bm{\xi}}\in\mathbb{R}^{p}:\|I^{1/2}_{{\bm{\theta}}_{0}}{\bm{\xi}}\|=c\\}$
for any $c>0$, while the second result tells us that if the hypothesis is one-
sided, then a test based on the GWSR statistic $\mathbf{W}_{n}$ would achieve
maximum (local) power for any fixed direction ${\bm{\xi}}\in\mathbb{R}^{p}$.
Although tests based on $\mathbf{W}_{n}$ can be locally asymptotically
optimal, the optimal score function $J(\cdot)$ (in (3.6)) depends on the
underlying distribution $f$, and is in general unknown.

######  Example 3.2 (Gaussian case).

Suppose it is known that $f(\cdot)$ (in Theorem 3.9) follows a Gaussian
distribution $N(\mathbf{0}_{p},\Sigma)$ with unknown nonsingular covariance
matrix $\Sigma$. Then with $S\mathbf{H}\sim N(\mathbf{0},I_{p})$, Theorem 3.9
states that the optimal score function (in (3.6)) is
$J(\mathbf{x})=\Sigma^{-1/2}\mathbf{x}$. Although $\Sigma$ is unknown, it may
be estimated by the sample covariance matrix $\widehat{\Sigma}$. Define
$\widehat{J}(\mathbf{x})=\widehat{\Sigma}^{-1/2}\mathbf{x}$. Then,

$\mathbf{W}_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})J(R_{n}(\mathbf{X}_{i}))=\Sigma^{-1/2}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})\right)$

has the same asymptotic distribution as

$\widehat{\Sigma}^{-1/2}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})R_{n}(\mathbf{X}_{i})\right).$
(3.9)

This means that using the data-driven score function $\widehat{J}(\cdot)$ has
the same asymptotic distribution as using $J(\cdot)$, and thus the GWSR test
based on (3.9) also has the same local asymptotic optimal properties.

Generalizing the above example, if we can estimate optimal score function
$J(\cdot)=-\frac{\nabla f(\nabla\psi^{*}(\cdot))}{f(\nabla\psi^{*}(\cdot))}$
in Theorem 3.9 sufficiently well such that

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})J(R_{n}(\mathbf{X}_{i}))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{n}(\mathbf{X}_{i})\widehat{J}(R_{n}(\mathbf{X}_{i}))+o_{p}(1),$

then the GWSR test using $\widehat{J}$ will be locally asymptotically optimal
as described in Theorem 3.9.

However, in general, estimating the optimal $J(\cdot)=-\frac{\nabla
f(\nabla\psi^{*}(\cdot))}{f(\nabla\psi^{*}(\cdot))}$ faces two main obstacles,
which are both hard problems without additional information. First,
$\frac{\nabla f(\cdot)}{f(\cdot)}$ is unknown and may be estimated
empirically. Estimating the gradient of a log-density function is in principle
a difficult non-parametric problem [60]. A positive result was given in
Hyvärinen, [60], where the author proposed the score matching method which
minimizes an empirical version of $\mathbb{E}_{\mathbf{X}\sim
f}\left[\|\mathbf{s}_{\tau}(\mathbf{X})-\nabla\log
f(\mathbf{X})\|_{2}^{2}\right]$ over $\tau$, with
$\mathbf{s}_{\tau}:\mathbb{R}^{p}\to\mathbb{R}^{p}$ is parametrized by $\tau$.
In Song and Ermon, [98], $\mathbf{s}_{\tau}$ is taken as a neural network;
$\mathbf{s}_{\tau}$ can also be taken to belong to a parametric family [60].
Another obstacle is the estimation of $\nabla\psi^{*}(\cdot)$ (which is the
transport map from $\nu_{S}$ to $P_{{\bm{\theta}}_{0}}$). Though with enough
data from $P_{{\bm{\theta}}_{0}}$, we can estimate $\nabla\psi^{*}(\cdot)$ up
to arbitrary precision [26, 70], the convergence rate may still suffer from
the curse of dimensionality [26, 70].

##  4 Finite Sample Performance

In this section we examine the finite sample performance of our distribution-
free procedures. We first illustrate the ARE results in Section 3, and then
compare our proposals with existing methods in the literature that test for
different types of symmetries. We will use the identity score function
$J(\mathbf{x})=\mathbf{x}$ in this section.

###  4.1 Empirical Validation of the ARE Results

Consider testing $\mathcal{G}$-symmetry for the following distributions:

  1. (a)

A bivariate Epanechnikov distribution (see Theorem 3.5) with
$\sigma_{i}=\frac{1}{\sqrt{5}}$ and independent components, shifted by
$0.05\cdot\mathbf{1}_{2}$. Here we test for central and sign symmetry.

  2. (b)

A bivariate standard Gaussian distribution shifted by
$0.1\cdot\mathbf{1}_{2}$. Here we test central, sign and spherical symmetry.

We will consider two types of effective reference distributions: (i) Gaussian
ERD: $N(\mathbf{0},I_{2})$, and (ii) Uniform ERD: ${\rm Uniform}(-1,1)^{2}$.
For different types of symmetries, we use the reference distributions $\nu$
recommended in Remark 3.3:

  * •

Central symmetry with Uniform ERD (CU): $\nu\sim{\rm Uniform}(0,1)\times{\rm
Uniform}(-1,1)$.

  * •

Sign symmetry with Uniform ERD (SU): $\nu\sim{\rm Uniform}(0,1)^{2}$.

  * •

Central symmetry with Gaussian ERD (CG): $\nu\sim|N(0,1)|\times N(0,1)$.

  * •

Sign symmetry with Gaussian ERD (SG): $\nu\sim|N(0,1)|\times|N(0,1)|$.

  * •

Spherical symmetry with Gaussian ERD (SpG):
$\nu\sim\sqrt{\chi^{2}_{2}}\times\\{0\\}$.

Figure 4: Left: The power curves for different methods based on sample size
$n$ or $0.864n$ (indicated by the percentage inside brackets) from a shifted
Epanechnikov distribution. Right: The power curves of different methods based
on $n$ (or $0.95n$) samples from a shifted Gaussian distribution. Here, CG,
CU, SG, SU, and SpG correspond to our GWSR test for different symmetries with
different ERDs; the first letter denotes the symmetry being tested (C for
central, S for sign, and Sp for spherical) and the second letter denotes the
ERD adopted (G for $N(\mathbf{0},I_{p})$ and U for ${\rm Uniform}[-1,1]^{p}$).
“t2” denotes the Hotelling’s $T^{2}$ test.

Within each replication, the empirical reference distribution $\nu_{n}$ is
obtained by drawing $n$ i.i.d. observations from $\nu$ and fixing it. The
empirical power (over $10^{4}$ replications) of our GWSR test at level
$\alpha=0.05$ and rejection region

$\mathbf{W}_{n}^{\top}\Sigma_{\rm
ERD}^{-1}\mathbf{W}_{n}\geq\chi_{p}^{2}(\alpha)$

are shown in Figure 4 for different sample sizes $n$. The power of Hotelling’s
$T^{2}$ test is also given, with rejection region:
$T^{2}\geq\chi_{p}^{2}(\alpha)$; see (3.5).

From Theorem 3.5 we know that the Epanechnikov distribution in (a) achieves
the equality in the 0.864 lower bound for Uniform ERD. Hence with a Uniform
ERD and $n$ samples, our method using central symmetry (CU), and sign symmetry
(SU) achieve almost the same power as Hotelling’s $T^{2}$ test using $0.864n$
samples; see the yellow, orange and green curves in the left panel of Figure
4.

If the ERD is Gaussian, then from Theorem 3.5, the ARE of our methods with
respect to Hotelling’s $T^{2}$ is strictly larger than 1. This can also be
observed in the left panel of Figure 4 — see the blue and grey curves
(corresponding to CG and SG) which are strictly above the yellow curve
(corresponding to Hotelling’s $T^{2}$), all obtained from the same number of
samples (i.e., 86.4% $n$). Also, observe that the blue curve (SG) is slightly
above the grey curve (CG); this is due to the fact that sign symmetry imposes
more constraints than central symmetry (note that the Epanechnikov
distribution satisfies both symmetry assumptions).

For (b), the shifted Gaussian example, we know that the lower bound in ${\rm
ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n})\geq 1$ is attained if Gaussian
ERD is used (see Theorems 3.5 and 3.6). This can also be observed in the right
panel of Figure 4 where we see that the four curves — yellow, dark blue, grey
and blue — are on top of each other.

that our method using any of the central symmetry (CG), sign symmetry (SG),
and spherical symmetry (SpG) has almost the same power as Hotelling’s $T^{2}$
test using the same number of samples. If using our method with a Uniform ERD,
then ${\rm ARE}(\mathbf{W}_{n},\overline{\mathbf{X}}_{n})=3/\pi\approx 0.95$.
Hence, again from the right panel of Figure 4, our method with Uniform ERD
using either central symmetry (CU) or sign symmetry (SU) and $n$ samples has
almost the same power as Hotelling’s $T^{2}$ test using $0.95n$ samples.

###  4.2 Power Comparison

In this subsection, we compare our methods with existing procedures in the
literature that test for different types of symmetries. We will use the
recommended Gaussian ERD with identity score (see Remark 3.3). Apart from
$\nu_{n}$ obtained via random sampling of the $\mathbf{h}_{i}$’s from $\nu$,
as adopted earlier, we will also use a quasi-Monte Carlo sequence (the Halton
transformed sequence) to construct $\nu_{n}$ as explained below. The Halton
sequence [48] $\\{\mathbf{U}_{i}^{p}\\}_{i\geq 1}$ (resp.
$\\{{U}_{i}^{1}\\}_{i\geq 1}$) provides a discretization of ${\rm
Uniform}(0,1)^{p}$ (resp. ${\rm Uniform}(0,1)$). Let $F_{Z}(\cdot)$ be the
distribution function of $Z\sim N(0,1)$, $F_{|Z|}(\cdot)$ be the c.d.f. of
$|Z|\sim|N(0,1)|$, and $F_{\|\mathbf{Z}\|}$ be the c.d.f. of
$\|\mathbf{Z}\|\sim\|N(\mathbf{0},I_{p})\|$. The Halton transformed
$\nu_{n}$’s for central, sign and spherical symmetries are given by the
discrete uniform distribution over
$\big{(}F_{|Z|}^{-1}(\mathbf{U}_{i,1}^{p}),F_{Z}^{-1}(\mathbf{U}_{i,2}^{p}),\ldots,F_{Z}^{-1}(\mathbf{U}_{i,d}^{p})\big{)}$,
for $i=1,\ldots,n$, the discrete uniform distribution over
$\big{(}F_{|Z|}^{-1}(\mathbf{U}_{i,1}^{p}),F_{|Z|}^{-1}(\mathbf{U}_{i,2}^{p}),\ldots,F_{|Z|}^{-1}(\mathbf{U}_{i,p}^{p})\big{)}$,
for $i=1,\ldots,n$, and the discrete uniform distribution over
$\big{(}F_{\|\mathbf{Z}\|}^{-1}({U}_{i}^{1}),0,\ldots,0\big{)}$, for
$i=1,\ldots,n$, respectively.

For low-dimensional scenarios (i.e., $p=2$), we will use the asymptotic
$\chi^{2}$ critical values for our methods and also for Hotelling’s $T^{2}$.
In the high-dimensional settings (i.e., $p=50$), we will use the exact
distribution of $T^{2}$ (assuming Gaussianity) and our statistics (which are
distribution-free). In particular, for the GWSR test, given the reference
points $\mathbf{h}_{1},\ldots,\mathbf{h}_{n}$, we sample 1000 observations
from the null distribution of $\mathbf{W}_{n}$, i.e.,
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{i}\mathbf{h}_{i}$ where $S_{i}$ are i.i.d.
from ${\rm Uniform}(\mathcal{G})$, and reject the null hypothesis if the
observed value of $\|\mathbf{W}_{n}\|^{2}$ is greater than 95% of the
observations from the null distribution.

We consider the following data distributions to test central symmetry
(C1-C10), sign symmetry (S1-S10), and spherical symmetry (Sp1-Sp10):

  1. C1.

Multivariate Gaussian: $\mathbf{X}\sim N\left(\mathbf{0},\begin{pmatrix}2&1\\\
1&3\\\ \end{pmatrix}\right)+\lambda\cdot\mathbf{1}_{2}$; $n=200$.

  2. C2.

Multivariate $t$-distribution with 1 degree of freedom, location parameter
$\lambda\cdot\mathbf{1}_{2}$ and scale parameter $\begin{pmatrix}2&1\\\ 1&3\\\
\end{pmatrix}$; $n=200$.

  3. C3.

Uniform distribution over $[-1,0]^{2}\cup[0,1]^{2}$, plus
$\lambda\cdot\mathbf{1}_{2}$; $n=200$.

  4. C4.

Exponential distribution: $n=100$, $p=2$, ${X}_{1},{X}_{2}$ i.i.d. follow
${\rm Exp}(1)-1$. $\mathbf{X}$ has mean $\mathbf{0}_{p}$.

  5. C5.

Exponential distribution (larger sample): Same as above except that $n=200$.

  6. C6.

Chi-squared distribution: $n=100$, $p=2$, ${X}_{1},{X}_{2}$ i.i.d. follow
$\chi^{2}(1)-1$. $\mathbf{X}$ has mean $\mathbf{0}_{p}$.

  7. C7.

Pareto distribution: $n=200$, $p=2$, ${X}_{1},{X}_{2}$ i.i.d. follow ${\rm
Pareto}(1)-2$. $\mathbf{X}$ has median $\mathbf{0}_{p}$.

  8. C8.

${\rm AR}(1)$ under ${\rm H}_{0}$: $n=200$, $p=50$, ${X}_{1}=Z_{1}$,
$X_{i}=0.5X_{i-1}+Z_{i}$, $i=2,\ldots,p$, where $Z_{i}$ are i.i.d. $N(0,1)$.

  9. C9.

${\rm AR}(1)$ under ${\rm H}_{1}$: Same as above except that the distribution
of $\mathbf{X}$ is shifted by $0.15\cdot\mathbf{1}_{d}$.

  10. C10.

AR$(1)$ (${\rm H}_{1}$, heavy-tailed): $n=200$, $p=50$, $Z_{i}$ are i.i.d.
from the standard $t$-distribution with 1 degree of freedom. ${Y}_{1}=Z_{1}$,
$Y_{i}=0.5Y_{i-1}+Z_{i}$, $i=2,\ldots,p$, $X_{i}=Y_{i}+0.9$, $i=1,\ldots,p$.

  11. S1.

Multivariate Gaussian: $\mathbf{X}\sim N\left(\mathbf{0},\begin{pmatrix}2&0\\\
0&3\\\ \end{pmatrix}\right)+\lambda\cdot\mathbf{1}_{2}$; $n=200$.

  12. S2.

Multivariate $t$-distribution with 1 degree of freedom, location parameter
$\lambda\cdot\mathbf{1}_{2}$ and scale parameter $\begin{pmatrix}2&0\\\ 0&3\\\
\end{pmatrix}$; $n=200$.

  13. S3.

Uniform distribution over $[-1,1]^{2}$, plus $\lambda\cdot\mathbf{1}_{2}$;
$n=200$.

  14. S4.

Correlation: $n=200$, $p=2$, $\mathbf{X}\sim
N\left(\mathbf{0},\begin{pmatrix}2&1\\\ 1&3\\\ \end{pmatrix}\right)$.

  15. S5.

Exponential distribution: $n=100$, $p=2$, ${X}_{1},{X}_{2}$ i.i.d. follow
${\rm Exp}(1)-1$.

  16. S6.

Chi-squared distribution: $n=100$, $p=2$, $X_{1}$, $X_{2}$ i.i.d. follow
$\chi^{2}(1)-1$.

  17. S7.

Laplace: $n=200$, $p=2$, $X_{1}$, $X_{2}$ i.i.d. follow a Laplace distribution
with location parameter 0.2 and scale parameter 1.

  18. S8.

High-dimensional (${\rm H}_{0}$): $n=200$, $p=50$,
$Z_{1},\ldots,Z_{p}\overset{i.i.d.}{\sim}N(0,1)$, $X_{i}=\sin(i)Z_{i}$.

  19. S9.

High-dimensional (${\rm H}_{1}$): $n=200$, $p=50$,
$Z_{1},\ldots,Z_{p}\overset{i.i.d.}{\sim}N(0,1)$, $X_{i}=\sin(i)Z_{i}+0.003$.

  20. S10.

High-dimensional (${\rm H}_{1}$, heavy-tailed): $n=200$, $p=50$,
$X_{i}=\sin(i)Z_{i}+0.003$; here $\mathbf{Z}$ follows a multivariate
$t$-distribution with 1 degree of freedom, location parameter $\mathbf{0}_{p}$
and scale parameter $I_{p}$.

  21. Sp1.

Multivariate Gaussian: $\mathbf{X}\sim
N\left(\mathbf{0},I_{2}\right)+\lambda\cdot\mathbf{1}_{2}$; $n=200$.

  22. Sp2.

Multivariate $t$-distribution with 1 degree of freedom, location parameter
$\lambda\cdot\mathbf{1}_{2}$ and scale parameter $I_{2}$; $n=200$.

  23. Sp3.

Uniform distribution over the unit disk
$\\{\mathbf{x}\in\mathbb{R}^{2}:\|\mathbf{x}\|\leq 1\\}$, plus
$\lambda\cdot\mathbf{1}_{2}$; $n=200$.

  24. Sp4.

Elliptical: $n=200$, $p=2$, $\mathbf{X}=(2Z_{1},Z_{2})^{\top}$, where $Z_{1}$,
$Z_{2}$ i.i.d. $N(0,1)$.

  25. Sp5.

Correlation: $n=200$, $p=2$, $\mathbf{X}\sim
N\left(\mathbf{0},\begin{pmatrix}1&\rho\\\ \rho&1\\\ \end{pmatrix}\right)$,
where $\rho=0.6$.

  26. Sp6.

Non-spherical uniform distribution: $n=1000$, $p=2$, $\mathbf{X}$ follows the
uniform distribution over $[-1,1]^{2}$.

  27. Sp7.

Chi-squared distribution: $n=100$, $p=2$, $X_{1}$, $X_{2}$ i.i.d. follow
$\chi^{2}(1)-1$.

  28. Sp8.

High-dimensional (${\rm H}_{0}$): $n=200$, $p=50$, $\mathbf{X}\sim
N(\mathbf{0},I_{p})$.

  29. Sp9.

High-dimensional (${\rm H}_{1}$): $n=200$, $p=50$, $\mathbf{X}\sim
N(\mathbf{0},I_{p})+0.05\cdot\mathbf{1}_{p}$.

  30. Sp10.

High-dim (${\rm H}_{1}$, heavy-tailed): $n=200$, $p=50$, $\mathbf{X}$ follows
the multivariate $t$-distribution with 1 degree of freedom, location parameter
$0.05\cdot\mathbf{1}_{p}$ and scale parameter $I_{p}$.

For central symmetry (C1-C10), we compare our GWSR test (denoted by “OT-
Wilcox”), our generalized sign test (denoted by “OT-sign”) and Hotelling’s
$T^{2}$ test with the depth-based test proposed in Dyckerhoff et al., [29]
(denoted by “DLP”), and the test proposed in Einmahl and Gan, [32] (denoted by
“EG”) which compares empirical measures of opposite regions. For EG, we use
the same parameter settings as in Einmahl and Gan, [32, Section 3].

For sign symmetry (S1-S10), we compare our proposals — OT-Wilcox and OT-sign —
with Hotelling’s $T^{2}$ test, the sign-change version of spatial sign test
(denoted by “SS”) [83, Section 6.1.2] and the sign-change version of the
spatial signed-rank test (denoted by “SSR”) [83, Section 7.1]. The last two
tests are implemented in the R package “MNM” [82].

For spherical symmetry (Sp1-Sp10), we compare our methods with Baringhaus, [6]
(denoted by “LB”), a test for multivariate spherical symmetry that is
consistent against any fixed alternative. For LB [6], a function $h$ needs to
be specified. When $p=2$, we use the function $h(t)=\frac{t-1/4}{17/8-t}$ as
in Baringhaus, [6, Section 4], which yields a tractable asymptotic null
distribution.When $p=50$, we use the function
$h(t)=(1-2tw+w^{2})^{-\lambda}-1$ [6, Equation 3.10] with $\lambda=24$ and
$w=1/4$ as in Baringhaus, [6, Section 5], and the same first-order
approximation to the upper tail probability for computing the critical value,
as advocated in Baringhaus, [6, Equation 4.4]. We also compare our methods
with Henze et al., [52] (denoted by “HHM”), a test for spherical symmetry
based on the empirical characteristic function. We use the suggested settings
in Henze et al., [52, Section 5].999More specifically, we use the Kolmogorov-
Smirnov type statistic with the hyper-parameter $\mathcal{R}=2$. For $p=2$, we
use 8 rings and 9 grid points as in Henze et al., [52, Section 5]. For $p=50$,
we use 8 rings and 200 grid points).

The nominal size of all the testing procedures is set at $\alpha=0.05$. Tables
2, 3 and 4 show the empirical power of the different methods for testing
central, sign and spherical symmetries, respectively. For our methods — OT-
Wilcox and OT-sign — and Hotelling’s $T^{2}$, the empirical power is averaged
over $10^{4}$ replications. For other competing methods, the empirical power
is based on 1000 replications, as some of them are computationally intensive.

Table 2: Power of the competing methods for testing central symmetry, with
nominal level 0.05. For OT-based methods, two choices of $\nu_{n}$ are used
(random sampling/[Halton transformed]). Multivariate Gaussian Distribution
---
C1 | $T^{2}$ | OT-Wilcox | OT-sign | DLP [29] | EG [32]
$\lambda=0.0$ | 0.06 | 0.05/[0.04] | 0.06/[0.06] | 0.05 | 0.04
$\lambda=0.1$ | 0.15 | 0.15/[0.13] | 0.11/[0.11] | 0.06 | 0.14
$\lambda=0.2$ | 0.49 | 0.46/[0.42] | 0.29/[0.29] | 0.08 | 0.46
$\lambda=0.3$ | 0.84 | 0.82/[0.80] | 0.55/[0.54] | 0.12 | 0.85
$\lambda=0.4$ | 0.98 | 0.98/[0.97] | 0.78/[0.77] | 0.20 | 0.98
Multivariate $t$-Distribution
C2 | $T^{2}$ | OT-Wilcox | OT-sign | DLP [29] | EG [32]
$\lambda=0.0$ | 0.03 | 0.06/[0.05] | 0.05/[0.05] | 0.05 | 0.04
$\lambda=0.2$ | 0.03 | 0.15/[0.13] | 0.19/[0.18] | 0.07 | 0.21
$\lambda=0.4$ | 0.04 | 0.48/[0.48] | 0.54/[0.54] | 0.12 | 0.62
$\lambda=0.6$ | 0.08 | 0.81/[0.79] | 0.86/[0.87] | 0.26 | 0.94
$\lambda=0.8$ | 0.12 | 0.96/[0.96] | 0.98/[0.97] | 0.49 | 1.00
Multivariate Uniform Distribution
C3 | $T^{2}$ | OT-Wilcox | OT-sign | DLP [29] | EG [32]
$\lambda=0.00$ | 0.05 | 0.05/[0.04] | 0.05/[0.05] | 0.05 | 0.05
$\lambda=0.03$ | 0.10 | 0.22/[0.18] | 0.09/[0.09] | 0.08 | 0.13
$\lambda=0.06$ | 0.27 | 0.64/[0.57] | 0.19/[0.18] | 0.15 | 0.45
$\lambda=0.09$ | 0.55 | 0.91/[0.88] | 0.35/[0.34] | 0.24 | 0.80
$\lambda=0.12$ | 0.81 | 0.99/[0.98] | 0.54/[0.52] | 0.37 | 0.97
Other Alternatives
C4-C7 | $T^{2}$ | OT-Wilcox | OT-sign | DLP [29] | EG [32]
Exponential | 0.07 | 0.10/[0.09] | 0.63/[0.66] | 0.84 | 0.88
Exponential (L) | 0.06 | 0.15/[0.15] | 0.91/[0.93] | 1.00 | 1.00
Chi-squared | 0.09 | 0.21/[0.19] | 0.83/[0.88] | 0.99 | 1.00
Pareto | 0.84 | 0.04/[0.02] | 0.90/[0.96] | 1.00 | 1.00
High-Dimensional Setting
C8-C10 | $T^{2}$ | OT-Wilcox | OT-sign | DLP [29] | EG [32]
AR$(1)$ (${\rm H}_{0}$) | 0.05 | 0.05/[0.05] | 0.04/[0.04] | — | —
AR$(1)$ (${\rm H}_{1}$) | 0.97 | 0.41/[0.99] | 0.07/[0.16] | — | —
AR$(1)$ (${\rm H}_{1}$, heavy-tailed) | 0.35 | 0.24/[0.89] | 0.06/[0.23] | — | —
Table 3: Power of the competing methods for testing sign symmetry, with
nominal level 0.05. For OT-based methods, two choices of $\nu_{n}$ are used
(random sampling/[Halton transformed]). Multivariate Gaussian Distribution
---
S1 | $T^{2}$ | OT-Wilcox | OT-sign | SSR [83] | SS [83]
$\lambda=0.0$ | 0.06 | 0.05/[0.04] | 0.05/[0.05] | 0.05 | 0.05
$\lambda=0.1$ | 0.21 | 0.20/[0.18] | 0.14/[0.14] | 0.19 | 0.16
$\lambda=0.2$ | 0.64 | 0.62/[0.60] | 0.43/[0.43] | 0.61 | 0.52
$\lambda=0.3$ | 0.94 | 0.93/[0.93] | 0.79/[0.79] | 0.94 | 0.87
$\lambda=0.4$ | 1.00 | 1.00/[1.00] | 0.97/[0.97] | 1.00 | 0.99
Multivariate $t$ Distribution
S2 | $T^{2}$ | OT-Wilcox | OT-sign | SSR [83] | SS [83]
$\lambda=0.0$ | 0.03 | 0.06/[0.05] | 0.04/[0.04] | 0.05 | 0.05
$\lambda=0.1$ | 0.04 | 0.10/[0.09] | 0.12/[0.12] | 0.09 | 0.12
$\lambda=0.2$ | 0.05 | 0.22/[0.20] | 0.29/[0.29] | 0.24 | 0.34
$\lambda=0.3$ | 0.06 | 0.44/[0.43] | 0.58/[0.58] | 0.48 | 0.66
$\lambda=0.4$ | 0.06 | 0.66/[0.65] | 0.83/[0.83] | 0.72 | 0.89
Multivariate Uniform Distribution
S3 | $T^{2}$ | OT-Wilcox | OT-sign | SSR [83] | SS [83]
$\lambda=.000$ | 0.05 | 0.05/[0.04] | 0.05/[0.05] | 0.06 | 0.05
$\lambda=.025$ | 0.12 | 0.16/[0.14] | 0.07/[0.07] | 0.10 | 0.08
$\lambda=.050$ | 0.33 | 0.51/[0.46] | 0.13/[0.13] | 0.29 | 0.18
$\lambda=.075$ | 0.65 | 0.83/[0.80] | 0.25/[0.25] | 0.58 | 0.37
$\lambda=.100$ | 0.89 | 0.97/[0.96] | 0.42/[0.42] | 0.83 | 0.60
Other Alternatives
S4-S7 | $T^{2}$ | OT-Wilcox | OT-sign | SSR [83] | SS [83]
Correlation | 0.06 | 0.06/[0.05] | 0.05/[0.05] | 0.05 | 0.05
Exponential | 0.07 | 0.14/[0.12] | 0.93/[0.93] | 0.34 | 0.79
Chi-squared | 0.09 | 0.30/[0.28] | 1.00/[1.00] | 0.59 | 0.98
Laplace | 0.73 | 0.81/[0.79] | 0.92/[0.92] | 0.83 | 0.88
High-Dimensional Setting
S8-S10 | $T^{2}$ | OT-Wilcox | OT-sign | SSR [83] | SS [83]
High-dim (${\rm H}_{0}$) | 0.04 | 0.05/[0.05] | 0.05/[0.05] | 0.05 | 0.05
High-dim (${\rm H}_{1}$) | 0.65 | 0.30/[0.28] | 0.49/[0.49] | 0.63 | 0.62
High-dim (${\rm H}_{1}$, heavy-tailed) | 0.05 | 0.17/[0.13] | 0.28/[0.28] | 0.27 | 0.35
Table 4: Power of the competing methods for testing spherical symmetry, with
nominal level 0.05. For OT-based methods, two choices of $\nu_{n}$ are used
(random sampling/[Halton transformed]). Multivariate Gaussian Distribution
---
Sp1 | $T^{2}$ | OT-Wilcox | OT-sign | LB [6] | HHM [52]
$\lambda=0.00$ | 0.06 | 0.05/[0.05] | 0.05/[0.06] | 0.05 | 0.05
$\lambda=0.05$ | 0.15 | 0.14/[0.14] | 0.07/[0.06] | 0.08 | 0.10
$\lambda=0.10$ | 0.42 | 0.42/[0.42] | 0.12/[0.12] | 0.18 | 0.26
$\lambda=0.15$ | 0.77 | 0.76/[0.76] | 0.21/[0.22] | 0.38 | 0.54
$\lambda=0.20$ | 0.95 | 0.95/[0.95] | 0.36/[0.37] | 0.62 | 0.84
Multivariate $t$ Distribution
Sp2 | $T^{2}$ | OT-Wilcox | OT-sign | LB [6] | HHM [52]
$\lambda=0.0$ | 0.03 | 0.07/[0.06] | 0.05/[0.06] | 0.04 | 0.05
$\lambda=0.1$ | 0.05 | 0.16/[0.15] | 0.09/[0.09] | 0.21 | 0.19
$\lambda=0.2$ | 0.06 | 0.46/[0.45] | 0.20/[0.20] | 0.75 | 0.60
$\lambda=0.3$ | 0.07 | 0.79/[0.78] | 0.39/[0.39] | 0.97 | 0.94
$\lambda=0.4$ | 0.10 | 0.96/[0.96] | 0.62/[0.62] | 1.00 | 0.99
Multivariate Uniform Distribution
Sp3 | $T^{2}$ | OT-Wilcox | OT-sign | LB [6] | HHM [52]
$\lambda=.000$ | 0.06 | 0.05/[0.04] | 0.05/[0.05] | 0.04 | 0.05
$\lambda=.025$ | 0.14 | 0.21/[0.19] | 0.06/[0.06] | 0.04 | 0.09
$\lambda=.050$ | 0.40 | 0.61/[0.59] | 0.09/[0.09] | 0.05 | 0.25
$\lambda=.075$ | 0.75 | 0.90/[0.90] | 0.14/[0.15] | 0.07 | 0.55
$\lambda=.100$ | 0.96 | 0.99/[0.99] | 0.23/[0.23] | 0.11 | 0.83
Other Alternatives
Sp4-Sp7 | $T^{2}$ | OT-Wilcox | OT-sign | LB [6] | HHM [52]
Elliptical | 0.06 | 0.06/[0.06] | 0.05/[0.05] | 0.50 | 1.00
Correlation | 0.06 | 0.06/[0.06] | 0.05/[0.05] | 0.50 | 1.00
Non-spherical uniform | 0.05 | 0.05/[0.05] | 0.05/[0.05] | 0.07 | 0.05
Chi-squared | 0.09 | 0.27/[0.25] | 0.45/[0.45] | 1.00 | 1.00
High-Dimensional Setting
Sp8-Sp10 | $T^{2}$ | OT-Wilcox | OT-sign | LB [6] | HHM [52]
High-dim (${\rm H}_{0}$) | 0.04 | 0.05/[0.05] | 0.05/[0.05] | 0.00 | 0.04
High-dim (${\rm H}_{1}$) | 0.54 | 0.68/[0.68] | 0.05/[0.05] | 0.00 | 0.15
High-dim (${\rm H}_{1}$, heavy-tailed) | 0.05 | 0.35/[0.35] | 0.05/[0.05] | 0.00 | 0.11

The first three examples (C1-C3, S1-S3, Sp1-Sp3) are location shift models, a
class of alternatives where OT-Wilcox is powerful. When $\lambda=0$, these
distributions are (centrally/sign/spherically) symmetric, and it can be seen
from Tables 2-4 that all methods have the correct size around 0.05. When
$\lambda>0$, in the Gaussian cases (C1, S1, Sph1), Hotelling’s $T^{2}$ test
works the best, and OT-Wilcox is only slightly inferior to it, which is a
consequence of 3.2 (or Theorems 3.5, 3.6). In the $t$-distribution example
(C2, S2, Sp2), Hotelling’s $T^{2}$ test has almost no power, while OT-Wilcox
still maintains high power. The power of the OT-sign test is even higher than
that of OT-Wilcox in settings C2 and S2. In the light-tailed uniform
distribution case (settings C3, S3, Sp3), Hotelling’s $T^{2}$ regains some
power, but it is still inferior to OT-Wilcox, which achieves the highest power
among all comparing methods.

The above scenarios C1-C3, S1-S3 and Sp1-Sp3 show that our OT-Wilcox is a
nonparametric competitor of Hotelling’s $T^{2}$ test, and is powerful in
detecting location shifts of a symmetric distribution. However, our methods
may be less powerful against non-location shift alternatives, such as a skewed
alternative with mean or median 0. We illustrate this using examples C4-C7,
S4-S6, and Sph4-Sph7, where the power of OT-Wilcox is lower compared to OT-
sign and most of the other competing methods, though it is still higher than
Hotelling’s $T^{2}$ test in general. Note that OT-Wilcox being powerless for
scenarios S4 and Sp4-Sp6 may be explained by Theorem 3.2: when testing sign
and spherical symmetry, OT-Wilcox will not be consistent against centrally
symmetric alternatives because $\mathbb{E}[\nabla\psi(\mathbf{X})]=\mathbf{0}$
(with $J(\mathbf{x})=\mathbf{x}$), a consequence of
$\nabla\psi(-\mathbf{x})=-\nabla\psi(\mathbf{x})$.

C8, S8 and Sp8 correspond to high-dimensional settings under ${\rm H}_{0}$. It
can be seen that both our methods and Hotelling’s $T^{2}$ have size close to
0.05, as expected. Under ${\rm H}_{1}$ (i.e., settings C9, S9, Sp9), OT-Wilcox
has nontrivial power, but is not competitive compared to Hotelling’s $T^{2}$
(see settings C9 and S9). One issue here is that when the dimension $p$ is
high, it can be difficult for $\nu_{n}$ to approximate $\nu$ well. For
example, for central symmetry, $\nu$ is supported on $2^{p-1}$ orthants, but
at most $n$ of the orthants have point(s) from $\nu_{n}$. Thus, from setting
C8 in Table 2, OT-Wilcox with $\nu_{n}$ obtained from random sampling is seen
to be less powerful than $T^{2}$. This may be improved by using $\nu_{n}$
obtained from the quasi-Monte Carlo (Halton transformed) points, which greatly
boosts the power of OT-Wilcox (see C8 in Table 2). However, Halton transformed
$\nu_{n}$ does not lead to significant power improvements for sign symmetry
(S8), spherical symmetry (Sp8), or in any of the low-dimensional examples,
where $\nu$ is easier to approximate. In the heavy-tailed high-dimensional
settings (i.e., C10, S10, Sp10), OT-Wilcox is clearly more powerful (and
robust) than Hotelling’s $T^{2}$.

Let us now compare the performance of the other methods mentioned above. Note
that DLP [29] and EG [32] are designed for testing bivariate central symmetry,
so they cannot be used in high-dimensional settings. LB [6] is powerless in
high-dimensional scenarios.101010Here the first-order approximation to the
upper tail probability is no longer accurate — the infinite product in [6,
Equation 4.4], which is only practically computable for small $p$, becomes
astronomically large when $p=50$. HHM [52] achieves relatively good
performance for settings Sp4, Sp5, Sp7, but is less satisfactory for the high-
dimensional cases Sp8-Sp10. SSR [83] and SS [83] perform quite well in testing
sign symmetry, except for location shifts of the uniform distribution (i.e.,
setting S3), where they have even lower power than Hotelling’s $T^{2}$.

In example S7 the ARE of our generalized sign test w.r.t. Hotelling’s $T^{2}$
is 2 (see Table 1); Table 3 (setting S7) shows that OT-sign does have higher
power than Hotelling’s $T^{2}$.

##  5 Distribution-Free Confidence Sets

We say that $\mathbf{X}\sim\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$ is
$\mathcal{G}$-symmetric around $\bm{\theta}^{*}\in\mathbb{R}^{p}$ if
$\mathbf{X}-\bm{\theta}^{*}\overset{d}{=}Q(\mathbf{X}-\bm{\theta}^{*})$ for
all $Q\in\mathcal{G}$. Here $\bm{\theta}^{*}$ is called the center of symmetry
for the distribution $\mu$. As a direct consequence of our results, the GWSR
test (or the generalized sign test) can be used to obtain a distribution-free
confidence set for $\bm{\theta}^{*}$. The idea is just to invert the
collection of hypotheses $\mathrm{H}_{0,\bm{\theta}}$ defined as:

$\mathrm{H}_{0,\bm{\theta}}:(\mathbf{X}-\bm{\theta})\stackrel{{\scriptstyle
d}}{{=}}Q(\mathbf{X}-\bm{\theta}),\quad\mbox{ for all }\;Q\in\mathcal{G},$

for $\bm{\theta}\in\mathbb{R}^{p}$, using the distribution-free (see
Proposition 2.2) GWSR test with data
$\\{\mathbf{X}_{i}-\bm{\theta}\\}_{i=1}^{n}$ at level $\alpha\in(0,1)$. This
yields an exact $(1-\alpha)$ distribution-free confidence set for
$\bm{\theta}^{*}$:

$\mathcal{C}:=\left\\{\bm{\theta}\in\mathbb{R}^{p}:\mathrm{H}_{0,\bm{\theta}}\mbox{
is accepted using data }\\{\mathbf{X}_{i}-\bm{\theta}\\}_{i=1}^{n}\right\\},$

without making any distributional assumptions on
$\mathbf{X}\sim\mu\in\mathcal{P}_{\rm ac}(\mathbb{R}^{p})$ except the
assumption of $\mathcal{G}$-symmetry. For implementational purposes, we
recommend using a fine grid of $\bm{\theta}$’s to test
$\mathrm{H}_{0,\bm{\theta}}$ and approximate $\mathcal{C}$. For $p\geq 2$, an
alternative to ‘gridding’ (which may be computationally infeasible) is to test
the hypothesis of $\mathcal{G}$-symmetry $\mathrm{H}_{0,\bm{\theta}}$ at
$\bm{\theta}=\mathbf{X}_{i}$, for each $i=1,\ldots,n$, and then to consider
the (possibly larger) convex hull of the accepted $\bm{\theta}$’s.

Figure 5 shows the distribution-free 95% confidence sets (colored in yellow;
obtained via a fine gridding of $\bm{\theta}\in\mathbb{R}^{p}$) using the GWSR
test when $p=2$ with the Gaussian ERD and the identity score function, when:
(i) $\mathcal{G}=\\{-I,I\\}$ (corresponding to central symmetry), and (ii)
$\mathcal{G}=O(p)$ (corresponding to spherical symmetry); see Remark 3.3 for
our choice of $\nu$ (here we used a random sample from $\nu$ to obtain
$\nu_{n}$ as in Section 4.2). The first two rows show data points drawn from
the Gaussian distribution $N(0.5\cdot\mathbf{1}_{2},I_{2})$; the third row
depicts data points obtained from the multivariate $t$-distribution with 1
degree of freedom with location parameter $0.5\cdot\mathbf{1}_{2}$ and scale
parameter $I_{2}$. In each example, the true center of symmetry
$\bm{\theta}^{*}$ is marked by “$+$”.

It can be seen that our GWSR based confidence sets successfully cover the true
unknown center $\bm{\theta}^{*}$ in the depicted cases. Moreover, for testing
spherical symmetry with Gaussian data (see the first row of Figure 5), the two
confidence sets — our proposal based on the GWSR test and the one obtained
from Hotelling’s $T^{2}$ test — are almost indistinguishable even for sample
size $n=20$. For testing central symmetry (see the second row of Figure 5)
using Gaussian data we observe a similar phenomenon, although for $n=20$ the
confidence set based on the Wilcoxon signed-rank test has a more jagged
boundary.

When the Gaussian assumption is violated and we have heavy-tailed observations
(see the third row of Figure 5) the confidence sets obtained from the
Hotelling’s $T^{2}$ test get severely inflated. However, our distribution-free
confidence sets are still quite small and reliable.

## Acknowledgements

The authors would like to thank Nabarun Deb for many helpful discussions.

Figure 5: The confidence sets obtained by inverting OT-Wilcox (for central
symmetry and spherical symmetry) and the Hotelling’s $T^{2}$ test. The true
center of the underlying Gaussian distribution is marked by “$+$”. The data
distribution is either Gaussian (the first two rows) or the $t$-distribution
with 1 degree of freedom.

##  Appendix A Some Discussions

###  A.1 Regularity Conditions

In this section, we describe the regularity conditions that are assumed in
Theorem 3.4, in order to study the power performance against local contiguous
alternatives. We adopt the standard smooth parametric model assumptions that
guarantee local asymptotic normality (LAN) (see e.g., Van der Vaart, [99,
Chapter 7]). Consider the parametric family consisting of all locational
shifts of a symmetric distribution $f$, i.e.,
$\\{\mathcal{P}_{\bm{\theta}}\\}_{{\bm{\theta}}\in\mathbb{R}^{p}}=\\{f(\cdot-{\bm{\theta}}):{\bm{\theta}}\in\mathbb{R}^{p}\\}$.
Assume:

  1. 1.

The family $\\{\mathcal{P}_{\bm{\theta}}\\}_{{\bm{\theta}}\in\mathbb{R}^{p}}$
is quadratic mean differentiable (QMD) at ${\bm{\theta}}_{0}=\mathbf{0}$ (see
[68, Definition 12.2.1] for relevant definitions).

  2. 2.

$\mathbb{E}_{\mathbf{X}\sim\mathcal{P}_{{\bm{\theta}}_{0}}}\left(\|\eta(\mathbf{X},{\bm{\theta}}_{0})\|^{2}\right)<\infty$,
where $\eta(\cdot,{\bm{\theta}}):=\frac{\nabla
f(\cdot-{\bm{\theta}})}{f(\cdot-{\bm{\theta}})}$ is the score function. Also
assume that the Fisher information at ${\bm{\theta}}_{0}$, that is,
$I_{{\bm{\theta}}_{0}}:=\mathbb{E}_{\mathbf{X}\sim\mathcal{P}_{{\bm{\theta}}_{0}}}\left[\eta(\mathbf{X},{\bm{\theta}})\eta(\mathbf{X},{\bm{\theta}})^{\top}\right]$
exists and is invertible.

###  A.2 Generating a Random Sign for Spherical Symmetry

To show that the procedure described in Remark 3.1 in Section 3.3 generates a
sample from the conditional distribution of ${\rm Uniform(O}(p))$ given that
it maps $\mathbf{v}$ to $\mathbf{w}$, we note that a uniform distribution over
${\rm O}(p)$ can be generated in the following way [30, Example 4.1]:

  1. 1.

First draw $\mathbf{\omega}$ from the uniform distribution over the sphere
$\\{\mathbf{x}\in\mathbb{R}^{p}:\|\mathbf{x}\|=1\\}$.

  2. 2.

Then draw $p(p-1)$ i.i.d. standard normal random variables
$\varepsilon\in\mathbb{R}^{p(p-1)}$ independent of $\mathbf{\omega}$.

  3. 3.

Apply Gram-Schmidt orthogonalization to the columns of $[\mathbf{\omega}\
\varepsilon]$ in the order of the columns. This yields an orthogonal matrix
${\rm GS}([\mathbf{\omega}\ \varepsilon])$ following the uniform distribution
over ${\rm O}(p)$ [30, Example 4.1].

  4. 4.

Define $f(\mathbf{\omega},\varepsilon):={\rm GS}([\mathbf{\omega}\
\varepsilon])[\mathbf{v}\ V]^{\top}$ which also follows the uniform
distribution over ${\rm O}(p)$ (as $[\mathbf{v}\ V]\in{\rm O}(p)$), by the
definition of the Haar measure (invariant under group action).

Note that the set
$\operatorname*{\arg\\!\min}_{Q\in\mathcal{G}}\|Q^{\top}\mathbf{x}-\mathbf{h}\|^{2}$
is described by $\\{Q\in{\rm O}(p):\mathbf{w}=Q\mathbf{v}\\}$. If
$f(\omega,\varepsilon)\mathbf{v}=\mathbf{w}$, we know that this condition is
equivalent to ${\rm GS}([\mathbf{\omega}\
\varepsilon])\mathbf{e}_{1}=\mathbf{w}$, where
$\mathbf{e}_{1}=(1,0,\ldots,0)^{\top}$, and it is further equivalent to
$\omega=\mathbf{w}$. Therefore, by the independence of $\omega$ and
$\varepsilon$, given $\omega=\mathbf{w}$, the conditional distribution of
$f(\omega,\varepsilon)$ is given by the unconditional distribution of
$f(\mathbf{w},\varepsilon)$. Thus the validity of the procedure is shown. ∎

A second way to generate the conditional distribution. From

$\\{Q\in\mathrm{O}(p):\;\mathbf{w}=Q\mathbf{v}\\}=\\{\mathbf{w}\mathbf{v}^{\top}+WUV^{\top}:\;U\in\mathrm{O}(p-1)\\},$

where $V$ and $W$ are $p\times(p-1)$ matrices such that
$V^{\top}V=W^{\top}W=I_{p-1}$,
$V^{\top}\mathbf{v}=W^{\top}\mathbf{w}=\mathbf{0}$, we can first generate $U$
from $\mathrm{O}(p-1)$ and then $\mathbf{w}\mathbf{v}^{\top}+WUV^{\top}$ has
the desired conditional distribution. The validity of this procedure can be
shown as follows:

  1. 1.

Draw $p^{2}$ i.i.d. standard normal variables
$\varepsilon\in\mathbb{R}^{p\times p}$ to form the $p\times p$ matrix
$[\varepsilon_{ij}]$.

  2. 2.

Apply Gram-Schmidt orthogonalization to the columns of $[\varepsilon_{ij}]$ in
the order of the columns. This yields an orthogonal matrix ${\rm
GS}([\varepsilon_{ij}])$ that follows the uniform distribution over ${\rm
O}(p)$ [30, Example 4.1].

  3. 3.

Define $f(\varepsilon):=[\mathbf{w}\ W]{\rm
GS}([\varepsilon_{ij}])[\mathbf{v}\ V]^{\top}$ which also follows the uniform
distribution over ${\rm O}(p)$ by the invariance property of the Haar measure.

If $f(\varepsilon)\mathbf{v}=\mathbf{w}$, we know that this condition is
equivalent to saying that the first column of ${\rm GS}([\varepsilon_{ij}])$
is $\mathbf{e}_{1}$. From the Gram-Schmidt procedure we know that this is
further equivalent to the event $\\{\varepsilon_{j1}=0,{\ \rm for\
}j=2,3,\ldots,p,\ {\rm and}\ \varepsilon_{11}>0\\}$ which only depends on the
first column of $[\varepsilon_{ij}]$. By independence, given this condition,
other $\varepsilon_{ij}$ from the second column to the last column are still
i.i.d. standard normal. Hence this implies the bottom right $(p-1)\times(p-1)$
sub-matrix of ${\rm GS}([\varepsilon_{ij}])$ (denoted by $U$) conditionally
follows the uniform distribution over ${\rm O}(p-1)$, and ${\rm
GS}([\varepsilon_{ij}])=\begin{pmatrix}1&\mathbf{0}^{\top}\\\ \mathbf{0}&U\\\
\end{pmatrix}.$ Hence the conditional distribution of the uniform distribution
$f(\varepsilon)$ given $\\{f(\varepsilon)\mathbf{v}=\mathbf{w}\\}$ has the
distribution of $[\mathbf{w}\ W]\begin{pmatrix}1&\mathbf{0}^{\top}\\\
\mathbf{0}&U\\\ \end{pmatrix}[\mathbf{v}\
V]^{\top}=\mathbf{w}\mathbf{v}^{\top}+WUV^{\top}$ with $U\sim{\rm