# A new class of nonparametric tests for second-order stochastic dominance
based on the Lorenz P-P plot

Tommaso Lando<EMAIL_ADDRESS>Sirio Legramanti

###### Abstract

Given samples from two non-negative random variables, we propose a new class
of nonparametric tests for the null hypothesis that one random variable
dominates the other with respect to second-order stochastic dominance. These
tests are based on the Lorenz P-P plot (LPP), which is the composition between
the inverse unscaled Lorenz curve of one distribution and the unscaled Lorenz
curve of the other. The LPP exceeds the identity function if and only if the
dominance condition is violated, providing a rather simple method to construct
test statistics, given by functionals defined over the difference between the
identity and the LPP. We determine a stochastic upper bound for such test
statistics under the null hypothesis, and derive its limit distribution, to be
approximated via bootstrap procedures. We also establish the asymptotic
validity of the tests under relatively mild conditions, allowing for both
dependent and independent samples. Finally, finite sample properties are
investigated through simulation studies.

###### keywords:

Bootstrap , Lorenz curve , Nonparametric test , Stochastic order

††journal: Elsevierlabel1label1footnotetext: Department of Economics,
University of Bergamo, Italylabel2label2footnotetext: Department of Finance,
VŠB-TU Ostrava, Czech Republic

##  1 Introduction

The theory of stochastic orders deals with the problem of comparing pairs of
random variables, or the corresponding distributions, with respect to concepts
such as size, variability (or riskiness), shape, ageing, or combinations of
these aspects. The main notion of stochastic ordering is generally referred to
as the usual stochastic order or first-order stochastic dominance (FSD), which
expresses the concept of one random variable being stochastically larger than
the other (Shaked and Shantikumar, 2007). For this reason, FSD has important
applications in all those fields in which “more” is preferable to “less”,
clearly including economics. However, FSD is a restrictive criterion, and is
rarely satisfied in real-world applications. This has pushed economic
theorists to develop finer concepts, which formed the theory of stochastic
dominance (SD), taking into account variability and shape aspects, in addition
to size (Hadar and Russell, 1969; Hanoch and Levy, 1969; Whitmore and Findlay,
1978; Fishburn, 1980; Muliere and Scarsini, 1989; Wang and Young, 1998). In
this regard, the most commonly used SD relation is the second-order SD (SSD),
expressing a preference for the random variable which is stochastically
larger, or, at least, less risky, therefore combining size and dispersion into
a single preorder. This has clear applications in economics, finance,
operations research, reliability, and many other fields in which decision-
makers typically prefer larger or at least less uncertain outcomes.

Given a pair of random samples from two unknown random variables of interest,
statistical methods may be employed to establish whether such variables are
stochastically ordered. In particular, we focus on a major problem in
nonparametric statistics, that is testing the null hypothesis of dominance
versus the alternative of non-dominance. With regard to SSD, some procedures
are available in the literature, some of which are described in the book of
Whang (2019). We will now recall a few of these approaches. Davidson and
Duclos (2000) proposed a test for SSD based on the distance between the
integrals of the cumulative distribution functions (CDF). The problem with
this test is that the dominance can be evaluated only on a fixed grid, which
may lead to inconsistency. Barrett and Donald (2003) employed a similar
approach, combined with bootstrap methods, to formulate a class of tests that
are consistent under the assumption that the distributions under analysis are
supported on a compact interval. Donald and Hsu (2016) leveraged a less
conservative approach to determine critical values, compared to Barrett and
Donald (2003), who relied on the least favourable configuration. Note that all
the aforementioned papers deal more generally with finite-order SD, then
obtaining SSD as a special case. Alternatively, other authors focused on tests
for the so-called Lorenz dominance, which is a scale-free version of SSD that
applies to non-negative random variables. For example, Barrett et al. (2014)
proposed a class of consistent tests for the Lorenz dominance that rely on the
distance between empirical Lorenz curves. In this case, critical values are
determined by approximating the limit distribution of a stochastic upper bound
of the test statistic, similar to Barrett and Donald (2003). Sun and Beare
(2021) used a different and less conservative bootstrap approach to improve
the power of such tests, and established asymptotic properties under less
restrictive distributional assumptions.

Some stochastic orders, including FSD, may be expressed and tested based on
the classic P-P plot, also referred to as the ordinal dominance curve (Beare
and Moon, 2015; Tang et al., 2017; Davidov and Herman, 2012; Beare and Clarke,
2022). Following a similar approach, we propose a new class of nonparametric
tests for SSD between non-negative random variables, in which the test
statistic is based on what we refer to as the Lorenz P-P plot (LPP), a kind of
P-P plot based on unscaled Lorenz curves. More precisely, the LPP is obtained
as the functional composition of the inverse unscaled Lorenz curve of one
distribution and the unscaled Lorenz curve of the other. The basic idea is
that, under SSD, the LPP does not exceed the identity function on the unit
interval. Therefore, the LPP stands out as a promising tool in detecting
deviations from the null hypothesis of SSD. Namely, any functional that
quantifies the positive part of the difference between the identity and the
LPP may be used to construct a test statistic. This gives rise to a class of
tests, according to the choice of the functional. The $p$-values can then be
computed via bootstrap procedures. In particular, we use a similar idea as in
Barrett et al. (2014) to asymptotically bound the size of the test, and
establish its consistency via the functional delta method.

The LPP may also be used to define families of fractional-degree orders that
are “between” FSD and SSD (and beyond), by using a simple transformation. In
this regard, we propose a method to define a continuum of SD relations, called
transformed SD, in the spirit of the recent works by Müller et al. (2017),
Lando and Bertoli-Barsotti (2020), and Huang et al. (2020). Interestingly, our
tests may be easily adapted to this more general family of orders, by simply
transforming the samples through the same transformation used in the
definition of transformed SD. In particular, FSD can be obtained as a limiting
case, in which the empirical LPP of the transformed sample tends to the
classic empirical P-P plot. This opens up the possibility of applying our
class of tests to a wide family of orders.

The paper is organised as follows. Section 2 introduces the LPP and describes
the idea behind the proposed family of tests. In Section 3, we propose an
estimator of the LPP and study its properties. The empirical process
associated with the LPP is investigated in Section 4, where we establish a
weak convergence result that may be used to derive asymptotic properties of
the tests. Namely, in Section 5 we establish bounded size under the null
hypothesis and consistency under the alternative one, which hold under both
independent and paired samples. The extension to a family of fractional-degree
orders is discussed in Section 6. In Section 7, we illustrate the finite
sample properties of the tests through simulation studies, focusing on tests
arising from sup-norm and integral-based functionals. Finally, Section 8
contains our concluding remarks. All the proofs are relegated in A.

##  2 Preliminaries

Throughout this paper, $H$ denotes a general CDF supported on the non-negative
half line, with finite mean $\mu_{H}$. In particular, we consider a pair of
non-negative random variables $X$ and $Y$ with CDFs $F$ and $G$, respectively,
and finite expectations. In the case when $F$ and $G$ are absolutely
continuous, we will denote their densities with $f$ and $g$, respectively.
Given that stochastic orders depend only on the distribution functions of the
random variables, for any order relation $\succ$ we may write $X\succ Y$ or
$F\succ G$, interchangeably.

Moreover, let $\ell^{\infty}(S)$ denote the class of bounded real-valued
functions defined on interval $S$ and equipped with the uniform norm
$||.||_{\infty}$, let $C[0,1]$ be the space of continuous real-valued
functions on [0,1] also equipped with the uniform norm, and $L^{1}(0,1)$ be
the class of integrable real-valued functions on the unit interval equipped
with the $L^{1}$ norm. Henceforth, “increasing” means “non-decreasing” and
“decreasing” means “non-increasing”. Given an increasing function $r$, we
denote with $r^{-1}(y)=\inf\\{x:r(x)>y\\}$ the right-continuos generalised
inverse of $r$. Finally, the symbol $\rightsquigarrow$ denotes weak
convergence as in Van der Vaart and Wellner (1996), while $\rightarrow_{p}$
denotes convergence in probability.

###  2.1 Stochastic dominance

We say that $X$ is larger than $Y$ with respect to FSD, denoted as
$X\geq_{1}Y$, if $F(x)\leq G(x),\forall x$. Equivalently, $X\geq_{1}Y$ if and
only if $\mathbb{E}u(X)\geq\mathbb{E}u(Y)$ for every increasing function $u$.
Within an economic framework, coherently with the expected-utility approach,
one may assume that $X$ and $Y$ represent monetary lotteries and $u$ is a
utility function. Under this perspective, FSD represents all non-satiable
decision-makers, that is all those with an increasing utility, and therefore
can be seen as one of the strongest ordering principles. On the other hand,
FSD has a limited range of applicability since, in real-world applications,
CDFs often cross and hence, most often, distributions cannot be ordered using
this criterion. For this reason, weaker ordering relations have been
introduced, among which the most important is the SSD. We say that $X$ is
larger than $Y$ with respect to SSD, denoted as $X\geq_{2}Y$, if
$\int_{-\infty}^{x}F(t)dt\leq\int_{-\infty}^{x}G(t)dt,\forall x$.
Equivalently, $X\geq_{2}Y$ if and only if $\mathbb{E}u(X)\geq\mathbb{E}u(Y)$
for every increasing and concave function $u$. In economics, SSD generally
represents all non-satiable and risk-averse decision-makers, expressing a
preference for the random variable with larger values or smaller dispersion.
For example, $X\geq_{2}Y$ entails that $\mathbb{E}X\geq\mathbb{E}Y$ and, in
case of equality, $\text{Var}(X)\leq\text{Var}(Y)$ and
$\Gamma(X)\leq\Gamma(Y)$, where $\Gamma$ denotes the Gini coefficient. The
above definitions may be generalised to $k$-th order SD, denoted as
$X\geq_{k}Y$, $k=1,2,3,...$, and represented by the following integral
inequality $F^{[k]}(x)\leq G^{[k]}(x),\forall x$, where $H^{[1]}=H$ and
$H^{[k]}(x)=\int_{-\infty}^{x}H^{[k-1]}(t)dt$, for $k\geq 2$.

Besides the classic definitions of SD discussed above, different notions —
often including FSD and SSD as special or limiting cases — have been studied
in the literature. Notable examples are the inverse SD (Muliere and Scarsini,
1989), which is based on recursive integration of the quantile function
instead of the CDF, and coincides with classic SD at degree 1 and 2, and also
some fractional-degree SD relations that interpolate FSD and SSD (see for
instance Müller et al. (2017)), as discussed in more detail in Section 6.

###  2.2 The Lorenz P-P plot

The goal of this paper is testing the null hypothesis
$\mathcal{H}_{0}:X\geq_{2}Y$ versus the alternative
$\mathcal{H}_{1}:X\not\geq_{2}Y$. This requires estimating some kind of
distance between the situation of dominance and the situation of non-
dominance. The classic solution (Davidson and Duclos, 2000; Barrett and
Donald, 2003) is to construct test statistics based on an empirical version of
the difference $\int_{-\infty}^{x}F(t)dt-\int_{-\infty}^{x}G(t)dt$, which is
expected to be large, at least at some point, if $\mathcal{H}_{0}$ is false.
However, the main issue with the usual definition of SSD, based on these
integrated CDFs, is that these are unbounded in $[0,\infty)$, so there are no
uniformly consistent estimators for such integrals unless both distributions
have bounded support. Not by chance, Barrett and Donald (2003) require that
$F$ and $G$ have a common bounded support $[0,a]$, with $a$ finite, to derive
consistent tests of stochastic dominance of order $k$, including SSD.

In order to avoid this limitation, we rely on an alternative but equivalent
definition of SSD, to be expressed in terms of the unscaled Lorenz curve,
which is always bounded. In particular, we observe that some stochastic orders
may be alternatively expressed in terms of a Q-Q plot (Lando et al., 2022) or
a P-P plot (Lehmann and Rojo, 1992). Similarly, SSD can be characterised by
the modified P-P plot described below.

Let $H^{-1}$ be the (left-continuous) quantile function of the CDF $H$. The
unscaled Lorenz curve of $H$ is defined as

$L_{H}(p)=\int_{0}^{p}H^{-1}(t)dt,\qquad p\in[0,1].$

The symbol $L_{H}$ is often used for the scaled version of the Lorenz curve,
that is $L_{H}/\mu_{H}$, while we use $L_{H}$ to denote the unscaled Lorenz
curve, for the sake of simplicity. Also note that
$L_{H}:[0,1]\rightarrow[0,\mu_{H}]$ is increasing, convex and continuous in
the unit interval. However, for technical reasons, we also let
$L_{H}(p)=+\infty$ for $p>1$. With this extension, the generalised inverse
function $L_{H}^{-1}:[0,\infty)\rightarrow[0,1]$ is increasing, concave, and
continuous in $[0,\mu_{H}]$, while $L_{H}^{-1}(y)=1$ for $y>\mu_{H}$. In the
case when $L_{H}$ is a scaled Lorenz curve, $L_{H}^{-1}$ is sometimes referred
to as the concentration curve (Goldie, 1977)

Now, given the pair of CDFs $F$ and $G$, consider the increasing continuous
function

$Z(p)=L_{G}^{-1}\circ L_{F}(p),\qquad p\in[0,1],$

which takes values in $[0,1\land L_{G}^{-1}(\mu_{F})]$, where $x\land y$
denotes the minimum between two generic real numbers $x$ and $y$. Letting
$\nu=1\land L_{F}^{-1}(\mu_{G})$, note that, if $\mu_{G}<\mu_{F}$, then
$\nu<1$ and $Z(p)=1$ for $p\in(\nu,1]$. Given some point $y=L_{F}(p)$, for
$p\in[0,1]$, the graph of $Z$ is a P-P plot with coordinates
$(L^{-1}_{F}(y),L^{-1}_{G}(y))$, which will be referred to as the Lorenz P-P
plot (LPP). Within an economic framework, $Z(p)$ returns the probability given
by $G$ to the average level of income corresponding to $L_{F}(p)$. In
particular, if such a level cannot be reached under $G$, we have $Z(p)=1$. The
LPP is scale-free, like the classic P-P plot, in particular, if $X$ and $Y$
are multiplied by a positive scale factor, then $Z$ remains unchanged.

It is well known (Shaked and Shantikumar, 2007, Ch. 4) that

$X\geq_{2}Y\iff L_{F}(p)\geq L_{G}(p),\ \forall p\in[0,1].$

Accordingly, it is easy to see that the above inequality can be equivalently
expressed in terms of $Z$:

$X\geq_{2}Y\iff Z(p)\geq p,\ \forall p\in[0,1].$ (1)

It is generally complicated to obtain an explicit expression of $Z$ for
parametric probabilistic models. Explicit calculations for the case of a
Weibull versus a unit exponential distribution are provided in Example 1
below, while a graphical illustration is given in Figure 1. Differently, and
more importantly for our testing purposes, the LPP can be computed quite
easily in the empirical case, as discussed in Section 3.

######  Example 1.

Consider the Weibull distribution $F(x)=1-\exp(-(x/b)^{a}),$ with $a,b>0$, and
the unit exponential distribution, $G(x)=1-\exp(-x)$, both supported on $x\geq
0$. In this case, the LPP has the following expression (see A for details):

$Z(p)=1\land\mathcal{R}\left[1-\exp\left({1+W_{-1}\left(\frac{b\left(\Gamma\left(1+{1}/{a}\right)-\Gamma\left(1+{1}/{a},-\log(1-p)\right)\right)-1}{e}\right)}\right)\right],$

where $\mathcal{R}$ indicates the real part of a complex number,
$\Gamma(\cdot,x)$ is the incomplete gamma function and $W_{-1}$ is the Lambert
function (Corless et al., 1996). Using the properties of SSD and the crossing
conditions described in Shaked and Shantikumar (2007), it is easy to verify
that $F\geq_{2}G$ if and only if $a\geq 1$ and $\mu_{F}=b\
\Gamma\left(1+{1}/{a}\right)\geq\mu_{G}=1$. Figure 1 shows the behaviour of
$Z$ when $F\geq_{2}G$ and $F\not\geq_{2}G$.

Figure 1: The LPP derived in Example 1 for: $a{=}2$, $b{=}1.5$ (dashed);
$a{=}2$, $b{=}0.8$ (dotted); $a{=}0.6$, $b{=}1.2$ (dot-dashed).

###  2.3 Detecting deviations from SSD

Denote the identity function by $I$. The representation of SSD in (1) can be
used to construct a test. In fact, $\mathcal{H}_{0}:X\geq_{2}Y$ is false if
and only if $I-Z$ is strictly positive at some point in the unit interval.
Accordingly, departures from SSD can be detected by quantifying the positive
part of the difference between $I$ and $Z$. This may be represented by some
functional $\mathcal{T}$ applied to the difference $I-Z$, with $\mathcal{T}$
satisfying some desirable properties. For instance, one may consider the
supremum of the difference between the identity and the LPP, thus yielding a
Kolmogorov-Smirnov type functional

$\mathcal{S}({I}-Z)=\sup_{p\in[0,1]}(p-Z(p)),$

which can also be expressed as
$\sup_{p\in[0,1]}(L_{F}^{-1}(p)-L_{G}^{-1}(p))$. Alternatively, we may
consider the area between $I$ and $Z$ where the first exceeds the latter,
namely,

$\mathcal{A}({I}-Z)=\int_{0}^{1}\max(0,p-Z(p))dp.$

Note that $\mathcal{S}$ and $\mathcal{A}$ completely characterise SSD, in that
$\mathcal{S}({I}-Z)=\mathcal{A}({I}-Z)=0$ if and only if $X\geq_{2}Y$, while
$\mathcal{S}({I}-Z)>0$ and $\mathcal{A}({I}-Z)>0$ if and only if
$X\not\geq_{2}Y$. More generally, one may consider a functional
$\mathcal{T}:C[0,1]\rightarrow\mathbb{R}$ with the following properties, where
$v_{1},v_{2}\in C[0,1]$:

  1. 1.

$\mathcal{T}(0)=0$;

  2. 2.

if $v_{1}(p)\leq 0,\forall p\in[0,1]$, then
${\mathcal{T}}(v_{2})\leq{\mathcal{T}}(v_{2}-v_{1})$;

  3. 3.

if $v_{1}(p)>0$ for some $p\in[0,1]$, then $\mathcal{T}(v_{1})>0$;

  4. 4.

$|\mathcal{T}(v_{1})-\mathcal{T}(v_{2})|\leq||v_{1}-v_{2}||_{\infty}$;

  5. 5.

for any positive constant $c>0$, $c\mathcal{T}(v_{1})=\mathcal{T}(cv_{1})$;

  6. 6.

$\mathcal{T}$ is convex.

Note that 1 and 4 imply that $\mathcal{T}$ is always bounded above by the
uniform norm. Verifying that $\mathcal{S}$ and $\mathcal{A}$ satisfy these
conditions is not difficult. These properties have been proposed and discussed
by Barrett et al. (2014), who introduced tests for the Lorenz dominance by
applying $\mathcal{T}$ to the difference between the (scaled) Lorenz curves,
that is $\mathcal{T}(L_{G}/\mu_{G}-L_{F}/\mu_{F})$. One may extend their
approach to SSD by considering $\mathcal{T}(L_{G}-L_{F})$. However, in this
paper, we propose leveraging $\mathcal{T}(I-Z)$. These two approaches measure
deviations from the null hypothesis differently. For instance, $I-Z$ is scale-
free by properties of the LPP, differently, if $X$ and $Y$ are multiplied by a
positive scale factor $c>0$, then the difference between the unscaled Lorenz
curve boils down to $c(L_{G}-L_{F})$. Moreover,
$-\max(\mu_{F},\mu_{G})<L_{G}-L_{F}<\max(\mu_{F},\mu_{G})$ whereas $-1<I-Z<1$,
so an advantage of using $I-Z$ is that its values are always within a bounded
interval.

######  Example 2.

To illustrate the behaviour of $I-Z$ versus that of $L_{G}-L_{F}$, we may take
$X$ and $Y$ as discrete random variables with uniform probabilities over the
supports $\\{13,20,40,50,100\\}$ and $\\{10,23,40,50,130\\}$, respectively. In
Figure 2, both approaches clearly show that $Y\not\geq_{2}X$. However, we also
have $X\not\geq_{2}Y$, which is the most difficult case to detect. This
situation is better highlighted by the graph of $I-Z$ compared to that of
$L_{G}-L_{F}$.

Figure 2: Crossing unscaled Lorenz curves.

##  3 Estimation of the LPP

###  3.1 Sampling assumptions

Let $\mathcal{X}=\\{X_{1},...,X_{n}\\}$ be an i.i.d. random sample from $F$,
and $\mathcal{Y}=\\{Y_{1},...,Y_{m}\\}$ an i.i.d. random sample from $G$. As
in Barrett et al. (2014), we will deal with two different sampling schemes:
independent sampling and matched pairs. In the independent sampling scheme,
the two samples $\mathcal{X}$ and $\mathcal{Y}$ are independent of each other,
while sample sizes $n$ and $m$ may differ. In contrast, in the matched-pairs
scheme, $n=m$ and we have $n$ i.i.d. pairs
$\\{(X_{1},Y_{1}),...,(X_{n},Y_{n})\\}$ drawn from a bivariate distribution
with $F$ and $G$ as marginal CDFs. For each sampling scheme, we will consider
the asymptotic regime in which $n\to\infty$,
$\lim_{n\rightarrow\infty}nm/(n+m)=\infty$ and
$\lim_{n\rightarrow\infty}n/(n+m)=\lambda\in[0,1]$. This assumption is quite
standard in the literature (Barrett and Donald, 2003) and implies that, as $n$
diverges, $m$ also goes to infinity with the same order.

###  3.2 Empirical LPP

The abovementioned random samples $\mathcal{X}$ and $\mathcal{Y}$ yield the
empirical CDFs

$F_{n}(x)=\frac{1}{n}\sum\nolimits_{i=1}^{n}\mathbbm{1}(X_{i}\leq x),\qquad
G_{m}(x)=\frac{1}{m}\sum\nolimits_{i=1}^{m}\mathbbm{1}(Y_{i}\leq x).$

We will denote the order statistics of rank $k$ from $\mathcal{X}$ and
$\mathcal{Y}$ with $X_{(k)}$ and $Y_{(k)}$, and the sample means by
$\overline{X}_{n}$ and $\overline{Y}_{m}$, respectively. By the plugin method,
the empirical counterparts of $L_{F}$ and $L_{G}^{-1}$ are just $L_{F_{n}}$
and $L^{-1}_{G_{m}}$, where $L_{F_{n}}(p)=\int_{0}^{p}F_{n}^{-1}(t)dt$ for
$p\in[0,1]$, $L_{G_{m}}$ is defined similarly, and $L_{G_{m}}^{-1}$ is the
inverse of $L_{G_{m}}$. As for the latter, coherently with our definition of
$L^{-1}_{G}$, we let $\widetilde{L}_{G_{m}}^{-1}(p)=1$ for
$p>\overline{Y}_{m}$. Note that $L_{F_{n}}$ coincides with the usual
definition of empirical unscaled Lorenz curve (Shorrocks, 1983), that is a
piecewise linear function joining the points
$(k/n,(1/n)\sum_{i=0}^{k}X_{(i)})$, $k=0,...,n$, with $X_{(0)}:=0$. These
definitions differ from the ones, based on step functions, considered by
Csörgö et al. (2013):

$\widetilde{L}_{F_{n}}(p)=\begin{cases}(1/n)\sum_{i=1}^{[np]+1}X_{(i)}\quad&p\in[0,1),\\\
\overline{X}_{n}&p=1,\\\ +\infty&p>1,\end{cases}$
$\widetilde{L}^{-1}_{G_{m}}(p)=\begin{cases}0\qquad&p\in[0,Y_{(1)}/m]\\\
(k-1)/m&p\in[(1/m)\sum_{i=1}^{k-1}Y_{(i)},(1/m)\sum_{i=1}^{k}Y_{(i)}),\quad
2\leq k\leq m\\\ 1&p\geq\overline{Y}_{m}\end{cases}$

where clearly
$\widetilde{L}_{G_{m}}^{-1}(p)=\inf\\{u:\widetilde{L}_{G_{m}}(u)>p\\}$. Note
that $L_{F_{n}}$ and $\widetilde{L}_{F_{n}}$ coincide at points $X_{(i)}$,
$i=1,..,n$, and, likewise, $L^{-1}_{G_{m}}$ and $\widetilde{L}^{-1}_{G_{m}}$
coincide at points $i/n$, so these alternative empirical versions of $L_{F}$
and $L_{G}^{-1}$ are clearly asymptotically equivalent.

According to the different empirical versions of $L_{F}$ and $L_{G}^{-1}$, we
may obtain two different estimators of $Z$. One may consider
${Z}_{n,m}={L}^{-1}_{G_{m}}\circ{L}_{F_{n}}$, which is a continuous piecewise
linear function. Alternatively, the composition
$\widetilde{Z}_{n,m}=\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}$ is
a step function with jumps in the points $i/n$, $i=1,...,n$, taking values
within the set $\\{j/m:j=1,...,m\\}$. Within an economic framework,
$\widetilde{L}_{G_{m}}^{-1}$ gives the relative frequency of those
observations from $Y$ whose level of income is at most
$(1/m)\sum_{j=1}^{k}Y_{(j)}$. Therefore, $\widetilde{Z}_{n,m}(k/n)$ returns
the relative frequency of those observations from $Y$ whose level of income is
at most $(1/n)\sum_{i=1}^{k}X_{(i)}$. Note that the value of ${Z}_{n,m}$ at
its “node” points $i/n$ does not generally coincide with the value of
$\widetilde{Z}_{n,m}$ at its jump points. Throughout this paper, we shall use
$\widetilde{Z}_{n,m}$ or ${Z}_{n,m}$ as is more convenient, since the two are
asymptotically equivalent. In fact, the sup-distance among
$\widetilde{Z}_{n,m}$ and $Z_{n,m}$ tends to zero as $n$ and $m$ tend to
infinity, as established in the following proposition.

######  Proposition 1.

As $n\to\infty$, $\sup|\widetilde{Z}_{n,m}-Z_{n,m}|\leq 1/m$. As a
consequence, if also $m\to\infty$, $\sup|\widetilde{Z}_{n,m}-Z_{n,m}|\to 0$.

Note that in our asymptotic scenario, whenever $n\to\infty$ we also have that
$m\to\infty$, hence Proposition 1 holds. Moreover, based on the strong uniform
consistency of Lorenz curve estimators and their inverse functions (Goldie,
1977; Csörgö et al., 2013), it is easy to prove the strong uniform consistency
of $Z_{n,m}$ and $\widetilde{Z}_{n,m}$.

######  Proposition 2.

As $n\to\infty$ and $m\to\infty$, $Z_{n,m}\rightarrow Z$ and
$\widetilde{Z}_{n,m}\rightarrow Z$ a.s. and uniformly in $[0,1]$.

##  4 Weak convergence of the LPP process

The empirical process associated with $Z$, henceforth referred to as the LPP
process, may be useful to characterise the limit distribution of the test
statistic under the null hypothesis of SSD. In this section, we study the
asymptotic properties of such a process. Define the LPP process as

$\mathcal{Z}_{n}(p)=\sqrt{r_{n}}(Z_{n,m}(p)-Z(p)),\qquad p\in[0,1],$

where ${r_{n}}={nm}/{(n+m)}$, and let $\nu_{n}=1\land
L_{F_{n}}^{-1}(\overline{Y}_{m})$ be the empirical counterpart of $\nu$. For
$\nu<1$ we know that $Z(t)=1$ when $t\in(\nu,1]$. In this case we have
$\sup|\mathcal{Z}_{n}\mathbbm{1}{(\nu,1]}|\rightarrow 0$ a.s., since also
$Z_{n,m}(p)=1$ for $t\in(\nu_{n},1]$ and $\nu_{n}\rightarrow\nu$ with
probability 1. In other words, the interval $(\nu_{n},1]$ contains no
information. Accordingly, we are particularly interested in the asymptotic
behaviour of $\mathcal{Z}_{n}$ restricted to the set $[0,\nu_{n}]$, namely,
$\mathcal{Z}_{n}\mathbbm{1}{[0,\nu_{n}]}$. Weak convergence of the LPP process
may be derived under the following assumptions.

######  Assumption 1.

Both $F$ and $G$ are continuously differentiable with strictly positive
density, and have finite moment of order $2+\epsilon$ for some $\epsilon>0$.
Moreover, $F(0)=G(0)=0$.

######  Assumption 2.

There exists some number $c>0$ such that $G^{-1}(0)=c$.

Note that Assumption 2 does not represent an effective limitation in terms of
applicability. In fact, if $G^{-1}(0)=0$, we can equivalently apply the test
to the shifted samples $\mathcal{X}+\epsilon$ and $\mathcal{Y}+\epsilon$, for
some arbitrarily small $\epsilon>0$, since $X\geq_{2}Y$ if and only if
$X+\epsilon\geq_{2}Y+\epsilon$ (this can be seen by writing SSD as
$\int_{0}^{p}(F^{-1}(t)+\epsilon)dt\geq\int_{0}^{p}(G^{-1}(t)+\epsilon)dt,\forall
p\in[0,1]$). In other words, SSD is translation-invariant. However, the
unscaled Lorenz curve, and therefore the LPP, is not translation-invariant, so
in practice, the result of the test may depend on $\epsilon$, as for large
values of $\epsilon$ the LPP may get closer to the identity. For this reason,
it is preferable to choose $\epsilon$ as small as possible (in our
computations, we set $\epsilon=10^{-4}$).

The following theorem establishes the weak convergence of $\mathcal{Z}_{n}$
under Assumptions 1–2, leveraging some recent results in Kaji (2018), which
enable the derivation of the the Hadamard differentiability of the map from
CDFs to quantile functions. As discussed in Sun and Beare (2021, Section 2.4),
this extends the applicability of earlier Hadamard differentiability
conditions, based on stronger distributional assumptions such as bounded
supports (Van der Vaart and Wellner, 1996, Lemma 3.9.23). Then, the weak
convergence of $\mathcal{Z}_{n}$ follows by the functional delta method (Van
der Vaart and Wellner, 1996, Sect. 3.9).

Henceforth, using the same notations as in Sun and Beare (2021), let
$\mathcal{B}$ be a centered Gaussian element of $C[0,1]\times C[0,1]$, with
covariance function

$Cov(\mathcal{B}(x_{1},y_{1}),\mathcal{B}(x_{2},y_{2}))=C(x_{1}\land
x_{2},y_{1}\land y_{2})-C(x_{1},y_{1})C(x_{2},y_{2}).$

Under the independent-sampling scheme, $C(x_{1},y_{1})=x_{1}y_{1}$ is the
product copula, whereas, under the matched-pairs scheme, $C$ is the copula
associated to the pair $(X_{i},Y_{i})$, $i=1,...,n$. Now, let
$\mathcal{B}_{1}(x_{1})=\mathcal{B}(x_{1},1)$ and
$\mathcal{B}_{2}(x_{2})=\mathcal{B}(1,x_{2})$. The random elements
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are Brownian bridges that are
independent under the independent-sampling scheme, but may be dependent under
the matched-pairs one.

######  Theorem 3.

Under Assumptions 1–2 and both independent-sampling and matched-pairs schemes,
we have:

$\sqrt{r_{n}}(Z_{n,m}-Z)\rightsquigarrow\mathcal{Z}\mathbbm{1}{[0,\nu]}\quad\text{in
}C[0,1],$

where

$\mathcal{Z}=\frac{-\sqrt{\lambda}\int_{0}^{Z}\mathcal{B}_{2}(p)dG^{-1}(p)+\sqrt{1-\lambda}\int_{0}^{.}\mathcal{B}_{1}dF^{-1}(p)}{G^{-1}\circ
Z}.$

It is interesting to observe that, if $X=_{d}Y$, the result of Theorem 3 boils
down to

$\displaystyle\sqrt{r_{n}}(Z_{n,m}(t)-t)$
$\displaystyle\rightsquigarrow\frac{1}{F^{-1}(t)}\int_{0}^{t}(\sqrt{\lambda}\mathcal{B}_{2}(u)-\sqrt{1-\lambda}\mathcal{B}_{1}(u))dF^{-1}(u)$
$\displaystyle=\frac{1}{F^{-1}(t)}\int_{0}^{t}\widetilde{\mathcal{B}}(u)dF^{-1}(u)=:\frac{\widetilde{\mathcal{L}}(t)}{F^{-1}(t)}\quad\text{in
}C[0,1],$

where $\widetilde{\mathcal{B}}$ is a Brownian bridge given by the linear
combination $-\sqrt{1-\lambda}\mathcal{B}_{1}+\sqrt{\lambda}\mathcal{B}_{2}$.
Finally, note that, by the asymptotic equivalence implied by Proposition 1,
all the results in this section still hold if one replaces $Z_{n,m}$ with
$\widetilde{Z}_{n,m}$.

##  5 Asymptotic properties of the test

As discussed in Section 2.3, deviations from $\mathcal{H}_{0}:X\geq_{2}Y$ can
be measured via the test statistic

$\mathcal{T}_{n}=\sqrt{r_{n}}\ {\mathcal{T}}(I-Z_{n,m}).$

Intuitively, we reject $\mathcal{H}_{0}$ if $\mathcal{T}_{n}$ is large enough.
However, since the null hypothesis is nonparametric, the main issue is how to
determine the null distribution of $\mathcal{T}_{n}$ — or, alternatively, of
an upper bound for $\mathcal{T}_{n}$ — under $\mathcal{H}_{0}$. Following the
approach of Barrett et al. (2014), it is easily seen that, under
$\mathcal{H}_{0}$, the test statistic $\sqrt{r_{n}}\ \mathcal{T}(I-Z_{n,m})$
is dominated by $\sqrt{r_{n}}\ \mathcal{T}(Z-Z_{n,m})$, which therefore can be
used to simulate $p$-values or critical values via bootstrap, thus ensuring
that the size of the test is asymptotically bounded by some arbitrarily small
probability $\alpha$. By the continuous mapping theorem, ${\sqrt{r_{n}}}\
\mathcal{T}(Z-Z_{n,m})$ is asymptotically distributed as
$\mathcal{T}(\mathcal{Z})$, allowing us to derive large-sample properties of
the test. The limit behaviour of $\mathcal{T}_{n}$ under the null and the
alternative hypotheses is established in the following lemma.

######  Lemma 4.

  1. 1.

Under $\mathcal{H}_{0}$, $\sqrt{r_{n}}\
\mathcal{T}(I-Z_{n,m})\leq\sqrt{r_{n}}\
\mathcal{T}(Z-Z_{n,m})\rightsquigarrow\mathcal{T}(\mathcal{Z})$. Moreover, for
any $\alpha<1/2$, the $(1-\alpha)$ quantile of the distribution of
$\mathcal{T}(\mathcal{Z})$ is positive, finite, and unique.

  2. 2.

Under $\mathcal{H}_{1}$, $\sqrt{r_{n}}\
\mathcal{T}(I-Z_{n,m})\rightarrow_{p}\infty$.

From a practical point of view, the limit distribution of
$\sqrt{r_{n}}\mathcal{T}(Z-Z_{n,m})$ under the null hypothesis may be
approximated using a bootstrap approach, as discussed in the next subsection.

###  5.1 Bootstrap decision rule

Let us denote the bootstrap estimators of the empirical CDFs $F_{n}$ and
$G_{m}$ as $F^{*}_{n}$ and $G^{*}_{m}$, respectively. These may be written as

$F^{*}_{n}(x)=\frac{1}{n}\sum\nolimits_{i=1}^{n}M_{i}^{1}\mathbbm{1}(x\leq
X_{i}),\qquad
G^{*}_{m}(x)=\frac{1}{m}\sum\nolimits_{i=1}^{m}M_{i}^{2}\mathbbm{1}(x\leq
Y_{i}),$

where $M^{1}=(M_{1}^{1},...,M^{1}_{n})$ and $M^{2}=(M_{1}^{2},...,M^{2}_{m})$
are independent of the data, and are drawn from a multinomial distribution
according to the chosen sampling scheme. In particular, under the independent-
sampling scheme, $M^{1}$ and $M^{2}$ are independently drawn from multinomial
distributions with uniform probabilities over $n$ and $m$ trials,
respectively. Under the matched-pairs scheme, we have $M^{1}=M^{2}$, which is
drawn from the multinomial distribution with uniform probabilities over $n=m$
trials, which means that we sample (with replacement) pairs of data, from the
$n$ pairs $\\{(X_{1},Y_{1}),...,(X_{n},Y_{n})\\}$. Correspondingly, by
applying the definitions discussed in Section 2, we obtain the bootstrap
estimators of the unscaled Lorenz curves, denoted with $L_{F^{*}_{n}}$ and
$L_{G^{*}_{n}}$, as well as the inverse $L^{-1}_{G^{*}_{n}}$. Then we may
define

$Z^{*}_{n,m}=L^{-1}_{G^{*}_{m}}\circ L_{F^{*}_{n}}.$

As is shown below, the random process $\sqrt{r_{n}}\
\mathcal{T}(Z_{n,m}-Z^{*}_{n,m})$ has the same limiting distribution as
$\mathcal{T}(\mathcal{Z})$. Therefore, bootstrap $p$-values are determined by

$p=P\\{\sqrt{r_{n}}\ \mathcal{T}(Z_{n,m}-Z^{*}_{n,m})>\sqrt{r_{n}}\
\mathcal{T}(I-Z_{n,m})\\},$

and can be approximated, based on $K$ simulations, by

$p\approx\frac{1}{K}\sum\nolimits_{k=1}^{K}\mathbbm{1}(\sqrt{r_{n}}\
\mathcal{T}(Z_{n,m}-Z^{*}_{k;n,m})>\sqrt{r_{n}}\ \mathcal{T}(I-Z_{n,m})),$

where $Z^{*}_{k;n,m}$ is the $k$-th resampled realisation of $Z^{*}_{n,m}$. As
usual, the test rejects $\mathcal{H}_{0}$ if $p<\alpha$.

The asymptotic behaviour of the test is addressed by the following
proposition.

######  Proposition 5.

Under Assumptions 1–2 and the sampling schemes in Section 3.1,

  1. 1.

If $\mathcal{H}_{0}$ is true, $\lim_{n\rightarrow\infty}P\\{\text{reject
}\mathcal{H}_{0}\\}\leq\alpha$;

  2. 2.

If $\mathcal{H}_{1}$ is true, $\lim_{n\rightarrow\infty}P\\{\text{reject
}\mathcal{H}_{0}\\}=1$

##  6 Extension to fractional-degree SD

An important topic in the SD theory is represented by SD relations between FSD
and SSD. This is motivated by the fact that FSD is a strong requirement, but,
on the other hand, SSD corresponds to total risk-aversion, which is a quite
restrictive principle in some cases (Müller et al., 2017). There are different
ways to define classes of orders which interpolate between FSD and SSD
(Fishburn, 1980; Leshno and Levy, 2002; Müller et al., 2017; Huang et al.,
2020; Lando and Bertoli-Barsotti, 2020). These different definitions lead to
different families of SD relations, typically parametrised by some real number
that represents the strength of the dominance. The first attempt in this
direction is ascribable to Fishburn (1980), who used fractional-degree
integration to interpolate the classic $k$-th order SD at all integer orders
$k\geq 1$. More recently, Müller et al. (2017), Huang et al. (2020), and Lando
and Bertoli-Barsotti (2020) proposed different parametrisations, with
different interpretations and properties, which coincide with classic SD only
at orders 1 and 2. In this section, we introduce a simple but very general
family of fractional-degree orders, which have the advantage that they can be
easily tested using the LPP method discussed earlier. Such a family can be
defined as follows.

Let $\mathcal{U}$ be the family of increasing absolutely continuous functions
over the non-negative half line. Under an economic perspective, $u$ may be
understood as a utility function, assigning values to monetary outcomes. For
some $u\in\mathcal{U}$, we say that $X$ dominates $Y$ with respect to
$u$-transformed stochastic dominance ($u$-TSD), and write $X\geq^{T}_{u}Y$, if
$u(X)\geq_{2}u(Y)$.

TSD has been already studied by Meyer (1977), who denotes it as SSD with
respect to a function $u$, and by Huang et al. (2020), who focus on a
particular parametric choice of $u$. In fact, since $u$-TSD represents SSD
between the transformed RVs $u(X)$ and $u(Y)$, then it can be simply expressed
through the LPP of $u(X)$ and $u(Y)$.

The behaviour of TSD clearly depends on the choice of $u$. To understand this
behaviour, let $u,\tilde{u}\in\mathcal{U}$ be two transformation functions
defined on the same interval. Generalizing a relevant definition of Chan et
al. (1990), we say that $u$ is more convex than $\tilde{u}$ and write
$u\geq_{c}\tilde{u}$ iff $u\circ\tilde{u}^{-1}$ is convex. The following
theorem shows that the definition of TSD may be equivalently expressed in
terms of expected utilities, thus generalizing Theorem 1 of Huang et al.
(2020).

######  Theorem 6.

$X\geq^{T}_{u}Y$ if and only if $\mathbb{E}(\phi(X))\geq\mathbb{E}(\phi(Y))$,
for every increasing utility $\phi$ such that $u\geq_{c}\phi.$

It is easy to see that, if $u$ and $\phi$ are twice differentiable, the
condition $u\geq_{c}\phi$ is equivalent to
$\rho_{\phi}(x)\geq\rho_{u}(x),\forall x$, where
$\rho_{g}(x)={g^{\prime\prime}(x)}/{g^{\prime}(x)}$ is the Arrow-Pratt index
of absolute risk aversion associated with the utility function $g$. Moreover,
the following general properties hold.

######  Theorem 7.

  1. 1.

If $u_{1}\geq_{c}u_{2}$ then $X\geq^{T}_{u_{1}}Y\implies X\geq^{T}_{u_{2}}Y$;

  2. 2.

$X\geq_{1}Y$ if and only if $X\geq^{T}_{u}Y,\forall u\in\mathcal{U}$.

Intuitively, the degree of convexity of the function $u$ determines the
strength of the SD relation, where SSD is obtained by taking $u$ to be the
identity function whereas FSD is obtained when $u$ is infinitely “steep”.

Families of utility functions within $\mathcal{U}$ can be obtained easily by
composing the quantile function and the CDF of two absolutely continuous RVs.
For example, one may consider the class of utility functions studied by Huang
et al. (2020) and given by $u_{c}(x)=\exp{((1/c-1)x)},$ for $c\in(0,1)$. Since
this paper deals with tests for non-negative random variables, we focus on a
simpler choice, that is $u_{\theta}(x)=x^{\theta}$, where $\theta\geq 0$.
Correspondingly, hereafter we reparametrise the ordering relation
$X\geq^{T}_{u_{\theta}}Y$ as $X\geq^{T}_{1+1/\theta}Y$, thus yielding a
continuum of SD relations which get stronger and stronger as $\theta$ grows.
By Theorem 7, this order is characterised by those utility functions which
have an Arrow-Pratt index larger than or equal to ${(\theta-1)}/{x}$. Since
$X\geq^{T}_{1+1/\theta}Y$ is equivalent to $X^{\theta}\geq_{2}Y^{\theta}$, a
test for $\mathcal{H}_{0}^{1+1/\theta}:X\geq^{T}_{1+1/\theta}Y$ versus
$\mathcal{H}_{1}^{1+1/\theta}:X\not\geq^{T}_{1+1/\theta}Y$ is readily obtained
by applying our method to the LPP of the transformed random samples
$\mathcal{X}^{\theta}$ and $\mathcal{Y}^{\theta}$. In particular, we consider
the generalised LPP, given by
$\widetilde{Z}_{n,m}^{\theta}=(\widetilde{L}^{\theta}_{G_{m}})^{-1}\circ\widetilde{L}^{\theta}_{F_{n}}$,
where $\widetilde{L}^{\theta}_{F_{n}}$ and $\widetilde{L}^{\theta}_{G_{m}}$
are the empirical (step-valued) unscaled Lorenz curves corresponding to the
transformed samples $\\{X_{i}^{\theta}:i=1,...,n\\}$, and
$\\{Y_{j}^{\theta}:j=1,...,m\\}$. $\widetilde{Z}_{n,m}^{\theta}$ is a
generalised P-P plot, in that it coincides with $\widetilde{Z}_{n,m}$ for
$\theta=1$. More interestingly, we prove that $\widetilde{Z}_{n,m}^{\theta}$
tends to the classic P-P plot of the non-transformed samples, that is,
$G_{m}\circ F_{n}^{-1}$ as $\theta\rightarrow\infty$, as depicted in Figure 3.
In particular, one may always find some $\theta$ large enough such that the
two P-P plots coincide, meaning that our tests may be also applied to FSD,
expressed as $G\circ F^{-1}(x)\geq x$ (tests for FSD based on the P-P plot
have been studied, for instance, by Davidov and Herman (2012) and Beare and
Clarke (2022)). In fact, this idea is coherent with the intuitive notion that,
for $\theta\rightarrow\infty$, the stochastic inequality
$X\geq^{T}_{1+1/\theta}Y$ reduces to $X\geq_{1}Y$, expressed as $F(x)\leq
G(x),\forall{x}$, as is formally established in the following theorem.

######  Theorem 8.

  1. 1.

For $\theta\rightarrow\infty$, $X\geq_{1}Y$ if and only if
$X\geq^{T}_{1+1/\theta}Y$.

  2. 2.

There exists some $\theta_{0}$ such that, for $\theta>\theta_{0}$, the
generalised LPP coincides with the classic P-P plot, that is,
$\widetilde{Z}_{n,m}^{\theta}=G_{m}\circ F_{n}^{-1}$.

Figure 3: The P-P plot (dashed) of two samples of size $n=m=20$ versus the
transformed LPP $\widetilde{Z}_{n,m}^{\theta}$ (solid), for $\theta=1,2,5,10$.
In this example, the plots coincide for $\theta\geq 48$.

To test FSD as a limit case of TSD, one should choose a value of $\theta$ that
ensures the result above. However, if $\theta$ is too large, computations may
be difficult, depending on the precision of the software used. We recommend
using $\theta=50$, which corresponds to testing $\geq^{T}_{1.02}$, for a good
approximation of FSD.

##  7 Simulations

We perform numerical analyses to investigate the finite-sample properties of
the proposed tests. In all simulations, we will consider a significance level
$\alpha=0.1$, and run $500$ experiments, with $500$ bootstrap replicates for
each experiment. For simplicity, we will set $n=m$, so henceforth we will drop
the subscript $m$. Namely, we will consider $n=m=50,100,200,500,1000$. All
computations have been performed in R.

In light of Proposition 1, instead of ${Z}_{n,m}$ we use $\widetilde{Z}_{n}$,
which can be computed faster. Accordingly, we consider two different test
statistics, namely $\mathcal{S}(I-\widetilde{Z}_{n})$ and
$\mathcal{A}(I-\widetilde{Z}_{n})$; see Section 2.3. For $n=m$, $\mathcal{S}$
and $\mathcal{A}$ can be rewritten, respectively, as

$\mathcal{S}(I-\widetilde{Z}_{n})=\max_{i}\left(\frac{i}{n}-\widetilde{Z}_{n}\left(\frac{i}{n}\right)\right),$

and

$\mathcal{A}(I-\widetilde{Z}_{n})=\sqrt{n}\frac{1}{n}\sum\nolimits_{i=1}^{n}\Psi\left(\frac{2i-1}{2n}\right),$

where $\Psi(t)=\max_{t}({0,t-\widetilde{Z}_{n}(t)})$. Our results are compared
with those obtained from the tests of Barrett and Donald (2003). These tests
are a natural competitor, as they represent the state of the art as far as
concerns SSD tests. In particular, Barrett and Donald (2003) propose three
bootstrap-based tests, based on a least favourable configuration, denoted as
KSB1, KSB2, and KSB3, which differ just for the bootstrap method employed to
simulate the $p$-values. We focus on KSB3 since it is based on the approach
that is most similar to ours, moreover, KSB3 seems to provide the best results
compared to KSB1 and KSB2 as far as concerns SSD, see tables II-A and II-B of
Barrett and Donald (2003). The $p$-values of KSB3 were computed using a grid
of evenly spaced values $0<t_{1}<t_{2}<...<t_{r}$, where $t_{r}$ is the
largest value in the pooled sample and $r=100$, as in Barrett and Donald
(2003). However, we did not notice substantial differences in increasing the
number of grid points.

Note that one pair of distributions gives rise to two different hypothesis
tests. In fact, one may test $\mathcal{H}_{0}:F\geq_{2}G$ versus
$\mathcal{H}_{1}:F\not\geq_{2}G$, but also the reversed hypothesis test,
denoted as $\mathcal{H}^{R}_{0}:G\geq_{2}F$ versus
$\mathcal{H}^{R}_{1}:G\not\geq_{2}F$. Except for the trivial case $F=G$, if
$Z$ does not cross the identity we may have that $\mathcal{H}_{0}$ is true
while $\mathcal{H}^{R}_{0}$ is false, or vice versa; differently, if $Z$
crosses the identity, $\mathcal{H}_{0}$ and $\mathcal{H}^{R}_{0}$ are both
false.

###  7.1 Size properties

To investigate the behaviour of the tests under the null hypothesis, we
simulate samples from the Weibull family, denoted by $W(a,b)$, with CDF
$F_{W}(x;a,b)=1-\exp\\{-\left({x}/{b}\right)^{a}\\}$. Since the mean of a
$W(a,b)$ is $b/q_{a}$, where $q_{a}=1/{\Gamma\left(1+\frac{1}{a}\right)}$, we
let $F\sim W(a,q_{a})$, for $a=1,1.25,1.5,1.75,2$, and fix $G\sim W(1,1)$. All
these distributions have mean 1, and in all such cases $\mathcal{H}_{0}$ holds
true. Clearly, for $a=1$ we have $F=G$ whereas the dominance becomes stronger,
and more apparent, for larger values of $a$.

The results in Tables 1, 2(a), 3(a) and 4(a) confirm that the proposed tests,
both with $\mathcal{S}$ and $\mathcal{A}$, behave as described in Proposition
5, part 1. Namely, the rejection frequence tends to be bounded by $\alpha=0.1$
when $\mathcal{H}_{0}$ is true. More specifically, we observe that the
rejection rate of the proposed tests tends to $\alpha$ in the case $F=G$ (see
Table 1), while it tends to 0 when the dominance holds strictly (see Tables
2(a), 3(a) and 4(a)). The rejection rate for the KSB3 test by Barrett and
Donald (2003) is also asymptotically bounded by $\alpha$ but, when the
dominance is stricter, it is still about $\alpha$ for $n=1000$. For such a
sample size, the rejection rate of both the proposed tests has already reached
0.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.09 | 0.15 | 0.10
100 | 0.10 | 0.15 | 0.11
200 | 0.12 | 0.15 | 0.11
500 | 0.09 | 0.10 | 0.09
1000 | 0.08 | 0.09 | 0.09
Table 1: Rejection rates under $\mathcal{H}_{0}$ (identical distribution):
$F=G\sim W(1,1)$

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.08 | 0.10 | 0.12
100 | 0.04 | 0.06 | 0.11
200 | 0.03 | 0.02 | 0.10
500 | 0.02 | 0.00 | 0.11
1000 | 0.00 | 0.00 | 0.09
(a) Test for $\mathcal{H}_{0}$ (true)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.19 | 0.31 | 0.12
100 | 0.22 | 0.29 | 0.11
200 | 0.27 | 0.36 | 0.11
500 | 0.50 | 0.54 | 0.12
1000 | 0.72 | 0.72 | 0.18
(b) Test for $\mathcal{H}_{0}^{R}$ (false)

Table 2: Rejection rates under $F\sim W(1.1,q_{1.1})$ and $G\sim W(1,1)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.04 | 0.05 | 0.12
100 | 0.04 | 0.02 | 0.13
200 | 0.02 | 0.00 | 0.12
500 | 0.01 | 0.00 | 0.11
1000 | 0.00 | 0.00 | 0.10
(a) Test for $\mathcal{H}_{0}$ (true)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.29 | 0.45 | 0.14
100 | 0.37 | 0.50 | 0.13
200 | 0.56 | 0.66 | 0.15
500 | 0.87 | 0.88 | 0.26
1000 | 0.99 | 0.99 | 0.48
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 3: Rejection rates under $F\sim W(1.2,q_{1.2})$ and $G\sim W(1,1)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.03 | 0.03 | 0.12
100 | 0.02 | 0.00 | 0.12
200 | 0.01 | 0.00 | 0.11
500 | 0.01 | 0.00 | 0.11
1000 | 0.00 | 0.00 | 0.08
(a) Test for $\mathcal{H}_{0}$ (true)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.41 | 0.57 | 0.16
100 | 0.56 | 0.67 | 0.18
200 | 0.81 | 0.85 | 0.28
500 | 0.99 | 0.99 | 0.51
1000 | 1.00 | 1.00 | 0.88
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 4: Rejection rates under $F\sim W(1.3,q_{1.3})$ and $G\sim W(1,1)$.

###  7.2 Power properties

We now investigate the behaviour of the tests under $\mathcal{H}_{1}$. In
particular, we focus on cases in which $F$ is dominated by $G$, so that
$\mathcal{H}_{0}$ should be rejected quite easily since $Z$ is always below
the identity. As we will discuss in 7.2.1, that the three tests considered
behave somewhat similarly in such cases. We also focus on critical cases, in
which neither of the two distributions dominates the other, and therefore $Z$
crosses the identity. In particular, the most critical situation for our class
of tests is the one in which $Z$ is above the identity except for a small
interval (see Figure 5). In some of the most difficult cases, the simulation
results summarized in 7.2.2 and 7.2.3 will show that $\mathcal{A}$ and KSB3
exhibit problems in rejecting $\mathcal{H}_{0}$, whereas the proposed
$\mathcal{S}$ test stands out as the most reliable one overall.

####  7.2.1 Weibull distribution

Using the same distributions as in Section 7.1, except for the case $F=G$, we
have that $F>_{2}G$ (strictly) and therefore $G\not\geq_{2}F$. In these cases,
$Z$ is always above the identity. The results, reported in Tables 2(b), 3(b)
and 4(b), show that the power of the tests increases with the sample size. In
particular, $\mathcal{A}$ seems to outperform $\mathcal{S}$ for smaller sample
sizes, while both the proposed $\mathcal{A}$ and $\mathcal{S}$ tests provide
larger power compared to the KSB3 test by Barrett and Donald (2003).

####  7.2.2 Lognormal mixture vs. lognormal distribution

As a second, more critical example, we focus on a special case considered by
Barrett and Donald (2003, Case 5). Here, $F$ is a mixture of lognormal
distributions, namely $F=0.9F_{LN}(.;0.85;0.4)+0.1F_{LN}(0.4,0.4)$, where
$F_{LN}(x;\mu,\sigma)=\Phi({(\log x-\mu)}/{\sigma})$ denotes the CDF of the
lognormal distribution, whereas $G=F_{LN}(.;0.86,0.6)$. These CDFs cross
multiple times and it can be seen that also $Z$ crosses the identity function
from below, so that $F\not\geq_{2}G$ but also $G\not\geq_{2}F$. In other
words, both $\mathcal{H}_{0}$ and $\mathcal{H}^{R}_{0}$ are false. In such
cases, the null hypothesis is hard to reject, because $Z$ crosses the identity
from above, in particular it exceeds the identity just in a small subset of
the unit interval. Note that Barrett and Donald (2003) just apply their test
to $\mathcal{H}_{0}$ versus $\mathcal{H}_{1}$, overlooking the reversed
situation, $\mathcal{H}^{R}_{0}$ versus $\mathcal{H}^{R}_{1}$.

As illustrated in Table 5, KSB3 seems to outperform our tests in detecting
$\mathcal{H}_{1}$. In particular, $\mathcal{A}$ exhibits quite a poor
performance with the sample sizes considered (to increase its power up to
0.68, we need to reach $n=5000$). Conversely, KSB3 has a really poor
performance in rejecting $\mathcal{H}^{R}_{0}$, while the proposed
$\mathcal{S}$ and $\mathcal{A}$ tests provide a large power in this critical
setting.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.25 | 0.17 | 0.43
100 | 0.43 | 0.15 | 0.59
200 | 0.58 | 0.10 | 0.74
500 | 0.89 | 0.08 | 0.98
1000 | 0.99 | 0.08 | 1.00
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.16 | 0.34 | 0.03
100 | 0.17 | 0.29 | 0.01
200 | 0.26 | 0.30 | 0.01
500 | 0.51 | 0.41 | 0.01
1000 | 0.84 | 0.68 | 0.02
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 5: Rejection rates under $F=0.9F_{LN}(.;0.85;0.4)+0.1F_{LN}(0.4,0.9)$,
$G=F_{LN}(.;0.86,0.6)$. In case a), the empirical power of $\mathcal{A}$
reaches 0.68 for $n=5000$.

####  7.2.3 Singh-Maddala Distribution

We now consider the Singh-Maddala distribution, denoted as $\text{SM}(a,q,b)$,
which has CDF $F_{\text{SM}}(x;a,q,b)=1-[1+(x/b)^{a}]^{-q}$. In all the
following scenarios, the scale parameter $b$ is set to 1, and hence omitted,
while the two shape parameters $a$ and $q$ vary.

As in Section 7.2.2, we generate scenarios in which $Z$ crosses the identity.
In particular, we target the worst-case scenarios for our proposed tests — in
order to investigate their limitations — by setting $F\sim\text{SM}(1.5,q)$
and $G\sim\text{SM}(1,q)$, for $q=1.2,1.5,1.8$. As shown in Figure 4, larger
values of $q$ correspond to cases in which it may be very hard to detect the
difference between $Z$ and the identity. Tables 6(a), 7(a) and 8(a) show that
KSB3 outperforms our tests in such critical cases. In particular, while the
performance of $\mathcal{S}$ significantly improves for larger samples and
lower $q$, the power of $\mathcal{A}$ is constantly close to 0, even for
$n=1000$ and $q=1.2$. However, the $p$-values for $\mathcal{A}$ tend to
decrease and to be less variable as $n$ grows, coherently with Proposition 5
(see Figure 5). This suggests that the power of $\mathcal{A}$ may eventually
tend to 1 for larger samples.

However, when applied to the reversed hypotheses $\mathcal{H}_{0}^{R}$ and
$\mathcal{H}_{1}^{R}$, the proposed $\mathcal{S}$ and $\mathcal{A}$ exhibit a
good performance, with rejection rates significantly increasing with $n$; see
Tables 6(b), 7(b) and 8(b). On the contrary, KSB3 has problems in detecting
non-dominance and its power remains always close to 0, even for large samples.

Figure 4: The behaviour of $Z$ for $q=1.2$ (solid), $q=1.5$ (dashed) and
$q=1.8$ (dotted) in the Singh-Maddala case. Especially for $q=1.8$, it becomes
very hard to detect deviations from $\mathcal{H}_{0}$. In the reversed cases,
the LPPs are just the inverse functions of these. Figure 5: Box-plots of the
simulated $p$-values of $\mathcal{A}$ in the most critical Singh-Maddala case
($q=1.8$).

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.02 | 0.00 | 0.39
100 | 0.05 | 0.00 | 0.52
200 | 0.05 | 0.00 | 0.65
500 | 0.11 | 0.00 | 0.87
1000 | 0.23 | 0.00 | 0.98
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.56 | 0.78 | 0.06
100 | 0.78 | 0.87 | 0.03
200 | 0.97 | 0.98 | 0.02
500 | 1.00 | 1.00 | 0.02
1000 | 1.00 | 1.00 | 0.03
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 6: Rejection rates under $F\sim\text{SM}(1.5,1.8)$,
$G\sim\text{SM}(1,1.8)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.11 | 0.03 | 0.57
100 | 0.10 | 0.01 | 0.66
200 | 0.19 | 0.00 | 0.84
500 | 0.48 | 0.00 | 0.98
1000 | 0.83 | 0.00 | 0.99
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.41 | 0.63 | 0.05
100 | 0.64 | 0.78 | 0.01
200 | 0.93 | 0.96 | 0.00
500 | 1.00 | 1.00 | 0.00
1000 | 1.00 | 1.00 | 0.00
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 7: Rejection rates under $F\sim\text{SM}(1.5,1.5)$,
$G\sim\text{SM}(1,1.5)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.19 | 0.06 | 0.71
100 | 0.25 | 0.03 | 0.78
200 | 0.47 | 0.02 | 0.91
500 | 0.83 | 0.01 | 0.99
1000 | 0.99 | 0.01 | 1.00
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.29 | 0.50 | 0.03
100 | 0.46 | 0.62 | 0.02
200 | 0.76 | 0.82 | 0.01
500 | 0.99 | 0.99 | 0.00
1000 | 1.00 | 1.00 | 0.00
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 8: Rejection rates under $F\sim\text{SM}(1.5,1.2)$,
$G\sim\text{SM}(1,1.2)$.

###  7.3 Paired samples

To simulate dependent samples we use the following method. First, we draw a
sample $\\{(Z^{1}_{i},Z^{2}_{i}):i=1,\ldots,n\\}$ from a bivariate normal
distribution, with standard marginals and correlation coefficient $\rho$.
Then, by transforming the data via the standard normal CDF $\Phi$, we obtain a
dependent sample from a bivariate distribution with uniform marginals
$\\{U_{i}^{1}=\Phi(Z^{1}_{i}):i=1,\ldots,n\\}$ and
$\\{U_{i}^{2}=\Phi(Z^{2}_{i}):i=1,\ldots,n\\}$. Finally, a dependent sample
from a bivariate distribution with margins $F$ and $G$ is obtained as
$\\{(F^{-1}(U_{i}^{1}),G^{-1}(U_{i}^{2})):i=1,\ldots,n\\}$. In particular, we
consider $\rho=0.25,0.5,0.75$. As in the previous subsections, we compare our
results with those of KSB3, although Barrett and Donald (2003) assume
independence to prove the consistency properties of the tests. Nevertheless,
our numerical findings reveal that KSB3 exhibits a good performance even in
the dependent case.

In this simulation study, we consider the same Singh-Maddala distributions as
in Section 7.2.3, focusing on the cases $q=1.2$ and $q=1.8$. The results,
reported in Tables 9–14, confirm the ones reported in the previous tables,
although it can be seen that a stronger dependence generally leads to larger
rejection rates. Even $\mathcal{A}$, which was struggling with independent
Singh-Maddala samples, shows a more evident decreasing trend in the empirical
distribution of the $p$-values when samples are paired (see Figure 6).

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.05 | 0.01 | 0.45
100 | 0.04 | 0.00 | 0.54
200 | 0.08 | 0.00 | 0.68
500 | 0.17 | 0.00 | 0.90
1000 | 0.32 | 0.00 | 0.99
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.64 | 0.84 | 0.07
100 | 0.87 | 0.94 | 0.03
200 | 1.00 | 1.00 | 0.02
500 | 1.00 | 1.00 | 0.01
1000 | 1.00 | 1.00 | 0.04
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 9: Rejection rates under $F\sim\text{SM}(1.5,1.8)$,
$G\sim\text{SM}(1,1.8)$, dependent samples $\rho=0.25$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.04 | 0.01 | 0.49
100 | 0.05 | 0.00 | 0.59
200 | 0.09 | 0.00 | 0.74
500 | 0.20 | 0.00 | 0.94
1000 | 0.44 | 0.00 | 1.00
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.75 | 0.93 | 0.07
100 | 0.94 | 0.99 | 0.04
200 | 1.00 | 1.00 | 0.02
500 | 1.00 | 1.00 | 0.02
1000 | 1.00 | 1.00 | 0.06
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 10: Rejection rates under $F\sim\text{SM}(1.5,1.8)$,
$G\sim\text{SM}(1,1.8)$, dependent samples $\rho=0.5$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.03 | 0.00 | 0.58
100 | 0.08 | 0.00 | 0.70
200 | 0.13 | 0.00 | 0.87
500 | 0.32 | 0.00 | 0.99
1000 | 0.72 | 0.00 | 1.00
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.92 | 0.99 | 0.07
100 | 1.00 | 1.00 | 0.04
200 | 1.00 | 1.00 | 0.04
500 | 1.00 | 1.00 | 0.05
1000 | 1.00 | 1.00 | 0.15
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 11: Rejection rates under $F\sim\text{SM}(1.5,1.8)$,
$G\sim\text{SM}(1,1.8)$, dependent samples $\rho=0.75$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.19 | 0.05 | 0.70
100 | 0.28 | 0.02 | 0.80
200 | 0.51 | 0.01 | 0.92
500 | 0.88 | 0.01 | 0.99
1000 | 1.00 | 0.02 | 0.99
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.34 | 0.58 | 0.03
100 | 0.56 | 0.71 | 0.01
200 | 0.89 | 0.91 | 0.01
500 | 1.00 | 1.00 | 0.00
1000 | 1.00 | 1.00 | 0.00
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 12: Rejection rates under $F\sim\text{SM}(1.5,1.2)$,
$G\sim\text{SM}(1,1.2)$, dependent samples $\rho=0.25$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.21 | 0.05 | 0.76
100 | 0.38 | 0.01 | 0.85
200 | 0.64 | 0.01 | 0.95
500 | 0.96 | 0.01 | 0.99
1000 | 1.00 | 0.03 | 0.99
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.42 | 0.68 | 0.03
100 | 0.69 | 0.84 | 0.00
200 | 0.96 | 0.97 | 0.00
500 | 1.00 | 1.00 | 0.00
1000 | 1.00 | 1.00 | 0.00
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 13: Rejection rates under $F\sim\text{SM}(1.5,1.2)$,
$G\sim\text{SM}(1,1.2)$, dependent samples $\rho=0.5$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.28 | 0.04 | 0.86
100 | 0.57 | 0.01 | 0.93
200 | 0.85 | 0.02 | 0.98
500 | 1.00 | 0.02 | 0.99
1000 | 1.00 | 0.09 | 1.00
(a) Test for $\mathcal{H}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.61 | 0.88 | 0.02
100 | 0.91 | 0.97 | 0.00
200 | 1.00 | 1.00 | 0.00
500 | 1.00 | 1.00 | 0.00
1000 | 1.00 | 1.00 | 0.00
(b) Test for $\mathcal{H}_{0}^{R}$ (false).

Table 14: Rejection rates under $F\sim\text{SM}(1.5,1.2)$,
$G\sim\text{SM}(1,1.2)$, dependent samples $\rho=0.75$. Figure 6: Box-plots of
the simulated $p$-values of $\mathcal{A}$ in the case $q=1.2$, $\rho=0.75$.

###  7.4 Test for FSD

As discussed in Section 6, our methodology also allows to test TSD, including
an approximation of FSD, obtained as $\geq_{1+1/\theta}^{T}$, letting
$\theta\rightarrow\infty$. We then apply the method described in Section 6 to
the same Singh-Maddala distributions studied in Section 7.2.3. Since in these
cases SSD does not hold, we also have that, a fortiori, the FSD null
hypothesis, denoted as $\mathcal{H}_{0}^{1}:F\geq_{1}G$, is also false. This
hypothesis can be tested using a sufficiently large value of $\theta$, as
discussed in Section 6. In particular, we set $\theta=50$, which corresponds
to approximating the FSD null hypothesis, $\mathcal{H}_{0}^{1}$, with
$\mathcal{H}_{0}^{1.02}$. Our method is compared with the FSD version of the
KSB3, as described in Barrett and Donald (2003). In contrast to the KSB3 test
for SSD, this test does not need a bounded support in order to be consistent.
All the tests considered tend to provide a larger simulated power compared to
the SSD case. This is logical since FSD is more stringent than SSD, and
therefore — for the same pairs of distributions — it is easy to detect
violations of FSD rather than of SSD. The results in Tables 15-17 show that
KSB3 tends to provide larger power than our $\mathcal{S}$ and $\mathcal{A}$
tests under $\mathcal{H}^{1}_{1}:F\not\geq_{1}G$, whereas, under the reversed
alternative $(\mathcal{H}^{1}_{1})^{R}:G\not\geq_{1}F$, KSB3 has worse
performance, also showing an unexpected behaviour, in that its rejection rates
first increase and then decrease as $n$ grows.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.11 | 0.14 | 0.24
100 | 0.22 | 0.16 | 0.41
200 | 0.44 | 0.21 | 0.64
500 | 0.86 | 0.42 | 0.92
1000 | 1.00 | 0.82 | 0.95
(a) Test for $\mathcal{H}^{1}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.03 | 0.29 | 0.19
100 | 0.10 | 0.38 | 0.19
200 | 0.31 | 0.53 | 0.14
500 | 0.92 | 0.92 | 0.05
1000 | 1.00 | 1.00 | 0.00
(b) Test for $(\mathcal{H}^{1}_{0})^{R}$ (false).

Table 15: FSD test. Rejection rates under $F\sim\text{SM}(1.5,1.2)$,
$G\sim\text{SM}(1,1.2)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.08 | 0.09 | 0.15
100 | 0.11 | 0.06 | 0.24
200 | 0.20 | 0.05 | 0.39
500 | 0.64 | 0.10 | 0.85
1000 | 0.96 | 0.29 | 0.97
(a) Test for $\mathcal{H}^{1}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.08 | 0.44 | 0.32
100 | 0.19 | 0.56 | 0.44
200 | 0.52 | 0.80 | 0.52
500 | 0.99 | 0.99 | 0.39
1000 | 1.00 | 1.00 | 0.22
(b) Test for $(\mathcal{H}^{1}_{0})^{R}$ (false).

Table 16: FSD test. Rejection rates under $F\sim\text{SM}(1.5,1.5)$,
$G\sim\text{SM}(1,1.5)$.

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.03 | 0.02 | 0.07
100 | 0.04 | 0.02 | 0.13
200 | 0.09 | 0.01 | 0.24
500 | 0.36 | 0.01 | 0.60
1000 | 0.77 | 0.02 | 0.92
(a) Test for $\mathcal{H}^{1}_{0}$ (false)

$n$ | $\mathcal{S}$ | $\mathcal{A}$ | KSB3
---|---|---|---
50 | 0.10 | 0.59 | 0.47
100 | 0.30 | 0.70 | 0.63
200 | 0.74 | 0.91 | 0.78
500 | 0.99 | 1.00 | 0.78
1000 | 1.00 | 1.00 | 0.67
(b) Test for $(\mathcal{H}^{1}_{0})^{R}$ (false).

Table 17: FSD test. Rejection rates under $F\sim\text{SM}(1.5,1.8)$,
$G\sim\text{SM}(1,1.8)$.

##  8 Concluding remarks

In this paper, we have proposed leveraging the LPP as a new tool to detect
deviations from SSD in the case of non-negative random variables. Thanks to
the extension to TSD, the same approach can be used to test different SD
relations, including FSD as a limit case. The asymptotic properties in Section
5 and the numerical results in Section 7 show that our family of tests can be
a valid alternative to the established tests based on the difference between
integrated CDFs, such as the tests of Barrett and Donald (2003). In
particular, the proposed test denoted with $\mathcal{S}$ often outperforms the
KSB3 test by Barrett and Donald (2003), which seems to have lower power in
most of the cases analysed, with a remarkable gap in some such cases. Among
the two types of test statistics proposed, the simulations reveal that the
sup-based test $\mathcal{S}$ is also overall more reliable than the integral-
based $\mathcal{A}$, which has lower power in the most critical cases.
However, both tests may be useful, in fact, according to our numerical results
in Section 7.2.3 and Section 7.2.2, $\mathcal{S}$ performs better when $Z$
crosses the identity from above, whereas $\mathcal{A}$ provides higher power
than $\mathcal{S}$ when $Z$ crosses the identity from below. Therefore, in
applications, it could be useful to use both tests and compare the $p$-values.
It is also worth noting that our proposed tests seem to improve in terms of
power when the samples are dependent.

In general, the advantage of using the LPP instead of integrated CDFs is that
it can be approximated uniformly, which makes it possible to establish
asymptotic properties without requiring bounded support; moreover, the LPP has
a different sensitivity in detecting violations of SSD, compared to other
methods. Finally, the power of our tests may be improved further by combining
the same proposed test statistics with different and less conservative
bootstrap schemes. The latter represents an interesting direction for future
work.

##  Appendix A Proofs

###### Calculations of Example 1.

The unscaled Lorenz curve of $F$ is

$L_{F}(p)=b\left(\Gamma\left(1+\frac{1}{a}\right)-\Gamma\left(1+\frac{1}{a},-\log(1-p)\right)\right),$

while

$L_{G}(p)=p+(1-p)\log(1-p).$

It is well known (e.g. Goldie, 1977) that $L_{G}$ can be expressed as
$L_{G}(p)=M_{G}\circ G^{-1}(p)$, with

$M_{G}(x)=\int_{0}^{x}t\ dG(t)=1-e^{-x}(x+1),\qquad x\geq 0.$

Noting that $M_{G}(x)\leq\mu_{G}=1$ for any $x\geq 0$, this function can be
inverted using the Lambert $W_{-1}$ function (Corless et al., 1996), that is
$M_{G}^{-1}(t)=-1-W_{-1}\left((t-1)/e\right)$. Accordingly,

$L_{G}^{-1}(t)=G\circ
M_{G}^{-1}(t)=1-\exp\left({1+W_{-1}\left(\frac{t-1}{e}\right)}\right).$

Finally, by composition, we obtain the expression of $Z$ in Example 1. ∎

###### Proof of Proposition 1.

$\widetilde{Z}_{n,m}-Z_{n,m}$ can be expressed as

$\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}-{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}=(\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}-{\widetilde{L}}^{-1}_{G_{m}}\circ{L}_{F_{n}})+(\widetilde{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}-{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}).$

For the first summand, which is the difference between two step functions, we
have
$\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}(p)\geq{\widetilde{L}}^{-1}_{G_{m}}\circ{L}_{F_{n}}(p)$
for every $p\in[0,1]$, since $\widetilde{L}_{F_{n}}(p)\geq{L}_{F_{n}}(p)$ for
every $p\in[0,1]$. Moreover,
$\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}(k/n)={\widetilde{L}}^{-1}_{G_{m}}\circ{L}_{F_{n}}(k/n)$
for $k=0,...,n$, while, within each interval $((k-1)/n,k/n)$, the difference
$\widetilde{L}^{-1}_{G_{m}}\circ\widetilde{L}_{F_{n}}(p)-{\widetilde{L}}^{-1}_{G_{m}}\circ{L}_{F_{n}}(p)$
is bounded above the height of the jumps of $\widetilde{L}^{-1}_{G_{m}}$, that
is, $1/m$. For the latter summand,
$\widetilde{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}-{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}\in[-\frac{1}{m},0]$,
since clearly ${L}^{-1}_{G_{m}}\circ{L}_{F_{n}}$ is the linear interpolator of
the jump points of the step function
$\widetilde{L}^{-1}_{G_{m}}\circ{L}_{F_{n}}$. Hence, the result follows. ∎

###### Proof of Proposition 2.

As proved in Theorem 10.1 and Theorem 13.2 of Csörgö et al. (2013),
$\widetilde{L}_{G_{m}}^{-1}$ and $\widetilde{L}_{F_{n}}$ converge strongly and
uniformly to $L_{G}^{-1}$ and $L_{F}$, respectively. Since $L_{G}^{-1}$ is
uniformly continuous in $[0,\infty)$ and
$\sup_{p}|L_{F_{n}}(p)-L_{F}(p)|\rightarrow 0$ almost surely, we obtain that
$\sup_{p}|L_{G}^{-1}\circ\widetilde{L}_{F_{n}}(p)-L_{G}^{-1}\circ
L_{F}(p)|\rightarrow 0$ almost surely. Then, for every $p\in(0,1)$,

$\displaystyle|\widetilde{L}_{G_{n}}^{-1}\circ\widetilde{L}_{F_{n}}(p)-L_{G}^{-1}\circ
L_{F}(p)|$
$\displaystyle\leq|\widetilde{L}_{G_{m}}^{-1}\circ\widetilde{L}_{F_{n}}(p)-L_{G}^{-1}\circ\widetilde{L}_{F_{n}}(p)|+|L_{G}^{-1}\circ\widetilde{L}_{F_{n}}(p)-L_{G}^{-1}\circ
L_{F}(p)|$
$\displaystyle\leq\sup_{p}|\widetilde{L}^{-1}_{G_{m}}(p)-L^{-1}_{G}(p)|+\sup_{p}|L_{G}^{-1}\circ\widetilde{L}_{F_{n}}(p)-L_{G}^{-1}\circ
L_{F}(p)|.$

Since both terms in the right-hand side converge to 0 with probability 1, we
obtain that $\widetilde{Z}_{n,m}$ converges strongly and uniformly to $Z$ in
$[0,1]$. By Proposition 1, $|\widetilde{Z}_{n,m}-Z_{n,m}|\rightarrow 0$ for
$n\rightarrow\infty$ and $m\rightarrow\infty$, therefore the same property is
satisfied by ${Z}_{n,m}$. ∎

###### Proof of Theorem 3.

Let $\mathbb{L}$ be the space of maps $z:[0,\infty)\rightarrow\mathbb{R}$ with
$\lim_{x\rightarrow-\infty}z(x)=0$ and $\lim_{x\rightarrow\infty}z(x)=1$, and
the norm $||z||_{\mathbb{L}}=\max\\{||z||_{\infty},||1-z||_{1}\\}$. As shown
by Kaji (2018), under assumption i), the map $\phi(F)=F^{-1}$, from CDFs to
quantile functions, is Hadamard differentiable at $F$, tangentially to the set
$\mathbb{L}_{0}$ of continuous functions in $\mathbb{L}$, with derivative map

$\phi_{F}^{\prime}(z)=-(z\circ F^{-1})(F^{-1})^{\prime}.$

The linear map ${\psi}(F^{-1})=\int_{0}^{.}F^{-1}(t)dt$ coincides with its
Hadamard derivative. Accordingly, by the chain rule (Van der Vaart and
Wellner, 1996, Lemma 3.9.3), the composition map $\psi\circ\phi:F\rightarrow
L_{F}$ is also Hadamard differentiable at $F$ tangentially to
$\mathbb{L}_{0}$, with derivative

$(\psi\circ\phi)^{\prime}_{F}(z)=\psi^{\prime}_{\phi(F)}\circ\phi_{F}^{\prime}(z)=-\int_{0}^{.}z\circ
F^{-1}(p)dF^{-1}(p).$

Now, observe that

$\begin{pmatrix}\sqrt{n}(F_{n}-F)\\\
\sqrt{m}(G_{n}-G)\end{pmatrix}\rightsquigarrow\begin{pmatrix}\mathcal{B}_{1}\circ
F\\\ \mathcal{B}_{2}\circ G\end{pmatrix}\text{ in
}\mathbb{L}\times\mathbb{L},$

as shown in Lemma 5.1 of Sun and Beare (2021). Then, the functional delta
method (Van der Vaart and Wellner, 1996, Theorem 3.9.13) implies the joint
weak convergence

$\begin{pmatrix}\sqrt{n}(L_{F_{n}}-L_{F})\\\
\sqrt{m}(L_{G_{n}}-L_{G})\end{pmatrix}=\begin{pmatrix}\sqrt{n}(\psi\circ\phi({F_{n}})-\psi\circ\phi({F}))\\\
\sqrt{m}(\psi\circ\phi({G_{n}})-\psi\circ\phi({G}))\end{pmatrix}\rightsquigarrow\begin{pmatrix}(\psi\circ\phi)^{\prime}_{F}(\mathcal{B}_{1}\circ
F)\\\ (\psi\circ\phi)^{\prime}_{G}(\mathcal{B}_{2}\circ G)\end{pmatrix}\\\
=\begin{pmatrix}-\int_{0}^{.}\mathcal{B}_{1}(t)dF^{-1}(t)\\\
-\int_{0}^{.}\mathcal{B}_{2}(t)dG^{-1}(t)\end{pmatrix}=:\begin{pmatrix}\mathcal{L}_{F}\\\
\mathcal{L}_{G}\end{pmatrix}\text{ in }C[0,1]\times C[0,1],$ (2)

Now, consider the process $\sqrt{m}(L^{-1}_{G_{m}}(t)-L^{-1}_{G}(t))$, for
$t\in[0,\mu_{G}]$. The LC $L_{G}$ is increasing and continuous on $[0,1]$,
therefore the inverse function $L_{G}^{-1}$ is increasing and continuous on
$[0,\mu_{G}]$, moreover the derivative $L_{G}^{\prime}=G^{-1}$ is strictly
positive in the unit interval (note that assumption ii) entails that
$G^{-1}(0)=c>0$). Then, by the inverse map theorem (Van der Vaart and Wellner,
1996, Lemma 3.9.20) the map $\eta:L_{G}\rightarrow L^{-1}_{G}$ is Hadamard-
differentiable at $L_{G}$, tangentially to the set of bounded functions on
$[0,1]$, with derivative

$\eta^{\prime}_{L_{G}}(z)=-\frac{z\circ L_{G}^{-1}}{L_{G}^{\prime}\circ
L_{G}^{-1}}=-\frac{z\circ L_{G}^{-1}}{G^{-1}\circ L_{G}^{-1}}.$

Since $r_{n}/n\rightarrow 1-\lambda$ and $r_{n}/m\rightarrow\lambda$, by (2)
and the functional delta method, the above result implies

$\sqrt{r_{n}}\begin{pmatrix}L_{F_{n}}-L_{F}\\\
L_{G_{n}}^{-1}-L_{G}^{-1}\end{pmatrix}\rightsquigarrow\begin{pmatrix}\lambda\mathcal{L}_{F}\\\
(1-\lambda)\eta^{\prime}_{L_{G}}(\mathcal{L}_{G})\end{pmatrix}=:\begin{pmatrix}\lambda\mathcal{L}_{F}\\\
(1-\lambda)\mathcal{C}_{G}\end{pmatrix}\text{ in }C[0,1]\times C[0,\mu_{G}],$
(3)

where $\mathcal{C}_{G}$ is defined as

$\mathcal{C}_{G}(t)=\frac{\int_{0}^{L_{G}^{-1}(t)}\mathcal{B}_{2}(p)dG^{-1}(p)}{G^{-1}\circ
L_{G}^{-1}(t)},\qquad t\in[0,\mu_{G}].$

Now, consider the maps $\pi=\psi\circ\phi:F\rightarrow L_{F}$,
$\theta:h\rightarrow L_{G}^{-1}\circ h$ and the composition map
$\zeta:C[0,1]\times C[0,\mu_{G}]\rightarrow C[0,\nu]$ defined by
$\zeta(\pi,\theta)(x)=\theta\circ\pi(x)$. Recall that the Hadamard derivative
of $\theta$ is $\theta^{\prime}_{L_{F}}(\alpha)=((L_{G}^{-1})^{\prime}\circ
L_{F})\alpha$, and $L_{G}^{-1}$ is uniformly norm-bounded, since
$(L_{G}^{-1})^{\prime}\leq 1/c$, therefore we can apply Lemma 3.2.27 of Van
der Vaart and Wellner (1996), which establishes that $\zeta$ is Hadamard
differentiable at $(\pi,\theta)$, tangentially to the set $C[0,1]\times
UC[0,\mu_{G}]$, where $UC[0,\mu_{G}]$ is the family of uniformly continuous
functions on $[0,\mu_{G}]$, with derivative

$\zeta^{\prime}_{\pi,\theta}(\alpha,\beta)(x)=\beta\circ\pi(x)+\theta^{\prime}_{\pi(x)}(\alpha(x))=\beta\circ
L_{F}(x)+\theta^{\prime}_{L_{F}}(\alpha(x))=\beta\circ
L_{F}(x)+((L_{G}^{-1})^{\prime}\circ L_{F})\alpha(x).$

Now, since $Z=\zeta(L_{F},L^{-1}_{G})$, using (3), the functional delta method
and the Hadamard differentiability of the composition map $\zeta$ give

$\displaystyle\sqrt{r_{n}}(Z_{n,m}-Z)\mathbbm{1}{[0,\nu]}$
$\displaystyle=\sqrt{r_{n}}(\zeta(L_{F_{n}},L^{-1}_{G_{m}})-\zeta(L_{F},L^{-1}_{G}))\mathbbm{1}{[0,\nu]}$
$\displaystyle\rightsquigarrow\zeta^{\prime}_{L_{F},L^{-1}_{G}}(\lambda\mathcal{L}_{F},(1-\lambda)\mathcal{C}_{G})=\sqrt{\lambda}\mathcal{C}_{G}\circ
L_{F}+\sqrt{1-\lambda}\frac{\mathcal{L}_{F}}{G^{-1}\circ L_{G}^{-1}\circ
L_{F}}$
$\displaystyle=\frac{-\sqrt{1-\lambda}\int_{0}^{.}\mathcal{B}_{1}dF^{-1}(p)+\sqrt{\lambda}\int_{0}^{Z}\mathcal{B}_{2}(p)dG^{-1}(p)}{G^{-1}\circ
Z}\quad\text{in }C[0,\nu],$

which implies the statement since
$\mathcal{Z}_{n}\mathbbm{1}{(\nu,1]}\rightsquigarrow 0$. ∎

###### Proof of Lemma 4.

Bear in mind that $\mathcal{Z}\mathbbm{1}{[0,\nu]}$ is a mean-zero Gaussian
process since it is obtained by integrating and normalizing Gaussian
processes. Under $\mathcal{H}_{0}$, $p-Z(p)\leq 0,\forall p\in[0,1]$, hence

$\sqrt{r_{n}}\ \mathcal{T}(I-Z_{n,m})\leq\sqrt{r_{n}}\
\mathcal{T}(Z-Z_{n,m})=\mathcal{T}(\sqrt{r_{n}}(Z-Z_{n,m}))\rightsquigarrow\mathcal{T}(\mathcal{Z}),$

where the last step follows from the continuous mapping theorem, since the map
$\mathcal{T}(f)$ satisfies
$|\mathcal{T}(f)-\mathcal{T}(g)|\leq||f-g||_{\infty}$, where $f,g$ are
continuous functions on the unit interval, as proved in Lemma 2 of Barrett et
al. (2014). The $(1-\alpha)$ quantile of the distribution of
$\mathcal{T}(\mathcal{Z})$ is positive, finite, and unique because
$\mathcal{Z}$ is a mean zero Gaussian process, so the proof follows by the
same arguments used in the proof of Lemma 4 in Barrett et al. (2014). Since,
by Proposition 2, $Z_{n,m}$ converges strongly and uniformly to $Z$, under
$\mathcal{H}_{1}$ we have
$\mathcal{T}(I-Z_{n,m})\rightarrow_{p}\mathcal{T}(I-Z)>0$. Finally,
multiplying by $\sqrt{r_{n}}$, we obtain the second result. ∎

###### Proof of Proposition 5.

As proved in Lemma 5.2 of Sun and Beare (2021),

$\begin{pmatrix}\sqrt{n}({F^{*}_{n}}-{F_{n}})\\\
\sqrt{m}({G^{*}_{n}}-{G_{n}})\end{pmatrix}\overset{as*}{\underset{M}{\rightsquigarrow}}\begin{pmatrix}\mathcal{B}_{1}\circ
F\\\ \mathcal{B}_{2}\circ G\end{pmatrix}\text{ in
}\mathbb{L}\times\mathbb{L},$

where $\overset{as*}{\underset{M}{\rightsquigarrow}}$ denotes weak convergence
conditional on the data a.s., see (Kosorok, 2008, p.20). The proof of Theorem
3 establishes the Hadamard-differentiability of the maps
$\psi\circ\phi:F\rightarrow L_{F}$ and $\eta\circ\psi\circ\phi:G\rightarrow
L_{G}^{-1}$, so that the functional delta method for the bootstrap implies

$\sqrt{r_{n}}\begin{pmatrix}L_{F^{*}_{n}}-L_{F_{n}}\\\
L_{G^{*}_{n}}^{-1}-L_{G_{n}}^{-1}\end{pmatrix}\overset{P}{\underset{M}{\rightsquigarrow}}\begin{pmatrix}\lambda\mathcal{L}_{F}\\\
(1-\lambda)\mathcal{C}_{G}\end{pmatrix}\text{ in }C[0,1]\times C[0,\mu_{G}],$

where $\overset{P}{\underset{M}{\rightsquigarrow}}$ denotes weak convergence
conditional on the data in probability (see Kosorok, 2008, p.20). Using the
Hadamard differentiability of the composition map
$\zeta(L_{F},L_{G}^{-1})=L_{G}^{-1}\circ L_{F}$, the functional delta method
for the bootstrap implies
$\sqrt{r_{n}}(Z_{n,m}^{*}-Z_{n,m})\overset{P}{\underset{M}{\rightsquigarrow}}\mathcal{Z}$,
which entails that
$\mathcal{T}(\sqrt{r_{n}}(Z_{n,m}-Z^{*}_{n,m}))\overset{P}{\underset{M}{\rightsquigarrow}}\mathcal{T}(-\mathcal{Z})=_{d}\mathcal{T}(\mathcal{Z})$
by the continuous mapping theorem. The test rejects the null hypothesis if the
test statistic exceeds the bootstrap threshold value
$c^{*}_{n}(\alpha)=\inf\\{y:P(\sqrt{r_{n}}\
\mathcal{T}(Z_{n,m}-Z^{*}_{k;n,m})>y|{\mathcal{X},\mathcal{Y}})\leq\alpha\\}$.
But the weak convergence result implies $c_{n}^{*}(\alpha)\rightarrow
c(\alpha)=\inf\\{y:P(\mathcal{T}({\mathcal{Z}})>y)\leq\alpha\\}$ in
probability, so Lemma 4 yields the result. ∎

###### Proof of Theorem 6.

Integrating by substitution, it is easy to see that $X\geq^{T}_{u}Y$ if and
only if $u(X)\geq_{2}u(Y)$, because $P(u(X)\leq t)=F_{X}\circ u^{-1}(t)$, and
similarly for $Y$. Hence, by setting $\phi=g\circ u$, the proof follows from
the classic characterisation of SSD, since $\mathbb{E}(g\circ
u(X))\geq\mathbb{E}(g\circ u(Y))$, for every increasing concave function $g$.
∎

###### Proof of Theorem 7.

Point 1) follows from the fact that $u_{1}(X)\geq_{2}u_{1}(Y)$ implies
$\mathbb{E}(u_{2}\circ u_{1}^{-1}\circ
u_{1}(X))=\mathbb{E}(u_{2}(X))\geq\mathbb{E}(u_{2}(Y))$, because the
composition $u_{2}\circ u_{1}^{-1}$ is concave by construction. The “only if”
part of point 2) is trivial. The “if” part follows from the characterisation
of FSD, taking into account that the equivalent condition of Theorem 6, that
is, $\mathbb{E}(\phi(X))\geq\mathbb{E}(\phi(Y)),\forall\phi\leq_{c}u$, for
every $u\in\mathcal{U}$, implies that such an inequality holds just for every
increasing $\phi\in\mathcal{U}$. Since any increasing function may be
approximated by a sequence of functions in $\mathcal{U}$, we have
$X\geq_{1}Y$. ∎

###### Proof of Theorem 8.

  1. 1.

$X\geq_{1+1/\theta}^{T}Y$ con be expressed as

$\int_{-\infty}^{x}F(t)du(t)\leq\int_{-\infty}^{x}G(t)du(t),\qquad\forall x.$
(4)

Integrating by parts and by substitution, we obtain that, for both $H=F$ and
$H=G$,

$\mathcal{I}^{\theta}_{H}(x)=\int_{0}^{x}H(t)du_{\theta}(t)=u_{\theta}(x)H(x)-\int_{0}^{x}u_{\theta}(t)dH(t)=u_{\theta}(x)H(x)-\int_{0}^{H(x)}u_{\theta}\circ
H^{-1}(y)dy.$

Hence,

$\frac{\mathcal{I}^{\theta}_{H}(x)}{u_{\theta}(x)}=H(x)-\int_{0}^{H(x)}\frac{u_{\theta}\circ
H^{-1}(y)}{u_{\theta}(x)}dy=H(x)-\int_{0}^{H(x)}\left(\frac{H^{-1}(y)}{x}\right)^{\theta}dy\rightarrow
H(x),$

by the Lebesgue dominated convergence theorem and bearing in mind that
${H^{-1}(y)}/{x}\leq 1$ as $y\leq H(x)$. Now, it is readily seen that
$X\geq^{T}_{1+1/\theta}Y$ if and only if
${\mathcal{I}^{\theta}_{F}(x)}/{u_{\theta}(x)}\leq{\mathcal{I}^{\theta}_{G}(x)}/{u_{\theta}(x)}$
for every $x$, which implies the result.

  2. 2.

Let $x_{1},...,x_{n}$ and $y_{1},...,y_{m}$ be ordered realisations from $X$
and $Y$, respectively. By properties of the power function, there exists some
number $\theta_{0}$ such that, for $\theta>\theta_{0}$,
$(1/n)\sum_{k=1}^{i}x_{k}^{\theta}\in((1/m)\sum_{k=1}^{j-1}y_{k}^{\theta},(1/m)\sum_{k=1}^{j}y_{k}^{\theta})$
if and only if ${x_{i}}/n\in({y_{j-1}}/m,{y_{j}}/m)$. In fact,

$x_{i}\left(\frac{1}{n}(\sum_{k=1}^{i-1}\left(\tfrac{x_{k}}{x_{i}}\right)^{\theta}+1)\right)\in\left(y_{j-1}\left(\frac{1}{m}(\sum_{k=1}^{j-2}\left(\tfrac{y_{k}}{y_{j-1}}\right)^{\theta}+1)\right),y_{j}\left(\frac{1}{m}(\sum_{k=1}^{j-1}\left(\tfrac{y_{k}}{y_{j}}\right)^{\theta}+1)\right)\right).$

Accordingly, for any $i=1,...,n$ and $\theta>\theta_{0}$,
$\widetilde{Z}_{n,m}^{\theta}$ returns $j/m$ if
${x_{i}}/n\in({y_{j-1}}/m,{y_{j}}/m)$, which coincides with the P-P plot
$G_{m}\circ F_{n}^{-1}$.

∎

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