# On standardness and the non-estimability of certain functionals of a set

Alejandro Cholaquidis
Facultad de Ciencias  The authors gratefully acknowledge support from grant
FCE-1-2019-1-156054, ANII (Uruguay), grant PID2020-116587GB-I00 funded by
MCIN/AEI/10.13039/501100011033 and ED431C 2021/24 funded by Consellería de
Cultura, Educación e Universidade.  Universidad de la República (Uruguay)
and
Leonardo Moreno
Departamento de Métodos Cuantitativos  Facultad de Ciencias Económicas y de
Administración  Universidad de la República (Uruguay)
and
Beatriz Pateiro-López
CITMAga (Galician Center for Mathematical Research and Technology)
and Departamento de Estatística  Análise Matemática e Optimización
Universidade de Santiago de Compostela (Spain)

###### Abstract

Standardness is a popular assumption in the literature on set estimation. It
also appears in statistical approaches to topological data analysis, where it
is common to assume that the data were sampled from a probability measure that
satisfies the standard assumption. Relevant results in this field, such as
rates of convergence and confidence sets, depend on the standardness
parameter, which in practice may be unknown. In this paper, we review the
notion of standardness and its connection to other geometrical restrictions.
We prove the almost sure consistency of a plug-in type estimator for the so-
called standardness constant, already studied in the literature. We propose a
method to correct the bias of the plug-in estimator and corroborate our
theoretical findings through a small simulation study. We also show that it is
not possible to determine, based on a finite sample, whether a probability
measure satisfies the standard assumption.

Keywords: Set Estimation, Standardness, Estimability, Topological Data
Analysis

##  1 Introduction

Set estimation deals with the problem of estimating a set, or a functional of
the set, from a sample of random points. The set in question could represent a
wide range of objects, such as the support of a probability distribution, its
boundary or levels sets, among others.

In this context, shape restrictions refer to limitations or constraints placed
on the shapes of the sets being estimated. These restrictions can take many
forms, such as requiring that the sets be convex, star-shaped, or have some
other geometric property. The purpose of these restrictions is typically to
make the estimation problem easier or more tractable, by reducing the space of
possible sets that need to be considered. Standardness is one such shape
restriction that is commonly used in set estimation.

In its simplest version, a set $S\subset\mathbb{R}^{d}$ is standard with
respect to the Lebesgue measure $\mu$ if there exist $\delta>0$ and
$\lambda>0$, such that, for all $x\in S$ and $0<\epsilon<\lambda$, the closed
ball $B(x,\epsilon)$, with center $x$ and radius $\epsilon$, satisfies

$\mu(B(x,\epsilon)\cap S)\geq\delta\mu(B(x,\epsilon)).$ (1)

Thus, $S$ is standard with respect to $\mu$ if a constant fraction of each
small ball centered at $S$ is contained in $S$. This prevents the set for
having sharp outward peaks, as shown in Figure 1. We will use the term
_standardness constant_ to refer to the supremum of the values of $\delta$
that satisfy the standardness condition on $S$ for some $\lambda>0$.
Standardness can also be defined more generally by substituting the Lebesgue
measure on the left-hand side of (1) with another appropriate measure $\nu$.
We will employ this broader notion of standardness (formally defined in
Section 2) throughout this paper. Specifically, we focus on the case where
$\nu$ is a probability measure supported on $S$. Hence, standardness can be
viewed as both a geometric and probabilistic condition. We investigate the
problem of estimating the standardness constant based on a random sample of
independent observations drawn from $\nu$. Initially, we examine a plug-in
type estimator and demonstrate its almost sure consistency under very general
restrictions on $S$ and $\nu$. This estimator has previously been studied in
Fasy et al. (2014) within the context of Topological Data Analysis (TDA),
where its convergence in probability was established under stronger
assumptions. We conduct a small simulation study that provides compelling
evidence that the performance of this estimator is poor, even for large sample
sizes. Then, we propose a method to correct the bias of the plug-in estimator
and establish the almost sure consistency of this modified estimator. Our
simulation study results demonstrate that the proposed correction
significantly improves the performance of the estimator.

(a)

(b)

(c)

Figure 1: The sets shown in (a) and (b) are standard with respect to $\mu$.
The set in (c) is not standard with respect to $\mu$.

Even though the standardness constant can be consistently estimated, we prove
that it is not possible to determine from a sample whether a set is standard
or not. This finding falls within the realm of a well-known problem in
statistics, that addresses the impossibility of consistently estimating
certain functionals of a probability measure from a finite sample. A similar
problem was recently analyzed in Cholaquidis et al. (2023), where it is shown
that it is not possible to determine, based on a finite sample, whether the
reach of the support of a density is zero or not. Determining which
functionals can be consistently estimated is a long-standing problem in
statistics, with roots dating back to the pioneering works of Bahadur and
Savage Bahadur and Savage (1956) and Le Cam and Schwartz LeCam and Schwartz
(1960). Subsequent studies have attempted to address this problem, including
the work of Donoho Donoho (1988), as well as more recent contributions such as
Fraiman and Meloche (1999) and Romano (2004), among others.

The paper is organized as follows. In Section 2 we take a closer look at other
shape restrictions considered in the literature on set estimation, along with
their interdependent associations with standardness. In addition, we provide a
comprehensive review of how the concept of standardness has been employed in
the set estimation literature, as well as in other domains such as Topological
Data Analysis (TDA). In Section 3, we provide the formal definition of the
standardness constant and introduce an additional restriction and some results
that will be needed throughout the work. In Section 4 we present the
estimators and theoretical results. Some numerical experiments illustrating
our results are given in Section 5. In Section 6 we prove, among others, that
it is not possible to determine based on a finite sample whether a set is
standard. All the technical proofs are given in the Appendix.

##  2 Background and related work

In this section, we provide background information and review relevant prior
work that contextualizes our research. First, we introduce some notation that
will be used throughout the paper. Then, we discuss some of the most common
shape restrictions considered in the literature on set estimation, in order of
decreasing restrictiveness: convexity, positive reach, $r$-convexity,
$r$-rolling condition, and cone-convexity. We then present the formal
definition of standardness and review its connection to the aforementioned
shape restrictions. Additionally, we discuss the crucial role of standardness
in the context of set estimation and TDA.

###  2.1 Some notation

Given a set $S\subset\mathbb{R}^{d}$, we will denote by
${\textnormal{int}}(S)$, $\overline{S}$ and $\partial S$ the interior, closure
and boundary of $S$, respectively, with respect to the usual topology of
$\mathbb{R}^{d}$.

If $A\subset\mathbb{R}^{d}$ is a Borel set, then $\mu(A)$ will denote its
$d$-dimensional Lebesgue measure. We will denote by $B(x,\varepsilon)$ the
closed ball in $\mathbb{R}^{d}$, of radius $\varepsilon$, centered at $x$, and
by $\omega_{d}$ the $d$-dimensional Lebesgue measure of the unit ball in
$\mathbb{R}^{d}$. The parallel set of $S$ of radii $\varepsilon$ will be
denoted as $B(S,\varepsilon)$, that is
$B(S,\varepsilon)=\\{y\in{\mathbb{R}}^{d}:\ \inf_{x\in
S}\|y-x\|\leq\varepsilon\\}$. Given a measure $\nu$, we use the notation
$\nu\ll\mu$ to indicate that $\nu$ is absolutely continuous with respect to
the Lebesgue measure. The support of a probability measure $\nu$ on
$\mathbb{R}^{d}$ is denoted by ${\textnormal{supp}}(\nu)$. The Hausdorff
distance between two compact non-empty sets in $A,C\subset{\mathbb{R}}^{d}$ is
given by $d_{H}(A,C)=\inf\\{\epsilon>0:A\subset B(C,\varepsilon),\ C\subset
B(A,\varepsilon)\\}$.

###  2.2 Shape constraints beyond convexity

Convexity is a key concept that has been studied extensively in mathematics.
In the context of set estimation, convexity is perhaps the most classical
geometric restriction. Since the early works in the 1960s, many contributions
have been made in this field, with a focus on various aspects including
support estimation Korostelev and Tsybakov (2012); Dümbgen and Walther (1996),
volume and boundary measure estimation Bräker and Hsing (1998); Baldin and
Reiß (2016), or level set estimation Hartigan (1987); Polonik (1995). We refer
to Brunel (2018) for a recent review of methods of set estimation in the
convex case. Nonetheless, convexity can be limiting when dealing with problems
that involve more flexible shape constraints. To address this limitation, a
lot of research has focused on extending the concept of convexity to more
general shape constraints.

_Positive reach._ The concept of sets of positive reach was first introduced
by Federer in Federer (1959), as a more flexible alternative to the classical
geometric restriction of convexity. While in a convex set all points in the
complement admit a unique projection onto it, in a set with positive reach,
only the points of the $r$-parallel neighborhood of the set are required to
have a unique projection. For instance, only the set (c) in Figure 1 has
positive reach.

In addition to the definition of reach, Federer (1959) obtains a
generalization of Steiner’s theorem and demonstrates the existence of
curvature measures, as well as other significant results. Numerous authors
have further explored the properties of sets of positive reach, see Thäle
(2008) for a review. An application of the positive reach condition in set
estimation can be found in Cuevas et al. (2007), where the goal is the
estimation of the boundary measure of a set. Recently, Cholaquidis et al.
(2023) proposed a universally consistent estimator for the reach of a set.

_$r$ -convexity._ The notion of convexity can also be generalized by that of
$r$-convexity. A set $S$ is $r$-convex, with $r>0$, if any point of its
complement can be separated from it by an open ball of radius $r$. For
instance, out of the sets depicted in Figure 1, only set (c) satisfies the
assumption of $r$-convexity. One way to understand how the notion of
$r$-convexity generalizes that of convexity is through the characterization of
convex sets in terms of supporting hyperplanes. In the case of $r$-convex
sets, the supporting hyperplanes would be replaced by supporting balls of
radius $r$. We refer to Walther (1999) as the first application of this
condition on set estimation, specifically in the context of level set
estimation. It can be also found in Rodríguez Casal (2007); Pateiro-López and
Rodríguez-Casal (2013); Rodríguez-Casal and Saavedra-Nieves (2016), in the
context of the problem of estimating a set or its boundary from a random
sample of points.

Proposition 1 in Cuevas et al. (2012) proves that every set with reach $r>0$
is also $r$-convex. Conversely, Theorem 6 in Cuevas et al. (2012) proves that
if $S$ is a compact $r$-convex set in $\mathbb{R}^{2}$ fulfilling an
additional mild regularity condition, then $S$ has positive reach (not
necessarily $r$).

_Rolling condition._ Rolling-type conditions and their relationship with the
smoothness of the boundary of a set were studied in Walther (1999).

######  Definition 2.1.

A closed set $S\subset\mathbb{R}^{d}$ is said to fulfill the (outside)
$r$-rolling condition with $r>0$ if for all $s\in\partial S$, there exists
some $x\in S^{c}$, such that $B(x,r)\cap\partial S=\\{s\\}$.

An (inside) $r$-rolling condition could be similarly defined by imposing the
condition on $\overline{S^{c}}$. Such double rolling-type smoothness
assumption has been considered, for instance, in Rodríguez-Casal and Saavedra-
Nieves (2016) for the estimation of the support, in Arias-Castro and
Rodríguez-Casal (2017); Arias-Castro et al. (2019); Aaron et al. (2022), for
the estimation of functionals of a set, such as volume or surface area, and in
Rodríguez-Casal and Saavedra-Nieves (2022), for density level sets estimation.
There is a close connection between the $r$-rolling condition and
$r$-convexity. Proposition 2 in Cuevas et al. (2012) proves that the
$r$-convexity implies the (outside) $r$-rolling condition. In general, the
converse implication does not hold.

Also, if both $S$ and $\overline{S^{c}}$ satisfy the (outside) $r$-rolling
condition then, $S$ and $\overline{S^{c}}$ are both $r$-convex and $\partial
S$ has positive reach, see Walther (1999) and Pateiro-López and Rodríguez-
Casal (2009). Of the sets displayed in Figure 1, the set in (b) satisfies an
(interior) $r$-rolling condition, while the set in (c) satisfies an (outside)
$r$-rolling condition.

_Cone-convexity._ A set $S$ is said to fulfill the $\rho$-cone-convex
property, with $\rho\in(0,\pi]$, if any point $x\in\partial S$ is the vertex
of an open finite cone with opening angle $\rho$, which does not intersects
$S$. As with $r$-convexity, one could understand cone-convexity as a
generalization of convexity, by replacing in this case the supporting
hyperplanes by supporting cones. Cone-convex sets may have indentations, with
their level of sharpness being constrained by the angle $\rho$. For example,
only set (b) in Figure 1 fails to satisfy the cone-convexity condition. Note
that the cone-convexity is a generalization of the rolling-type conditions
commented above. If $S$ satisfies the (outside) $r$-rolling condition, then it
is $\rho$-cone-convex for $\rho<\pi/2$. The converse implication does not
hold. We refer to Cholaquidis et al. (2014) for more details on cone-convexity
and its application in set estimation.

_Standardness._ This restriction, as given in (1) with respect to the Lebesgue
measure, leads to a broader family of sets than that of convex sets. Regarding
the other above mentioned shape constraints, it can be easily seen that
standardness with respect to $\mu$ is implied by, but not equivalent to, an
(inside) $r$-rolling condition.

In a more general version of standardness, the Lebesgue measure in the left-
hand side of (1) can be replaced with another suitable measure.

######  Definition 2.2.

A Borel set $S\subset\mathbb{R}^{d}$ is said to be standard with respect to a
Borel measure $\nu$ if there exists $\lambda>0$ and $\delta>0$ such that, for
all $x\in S$,

$\nu(B(x,\varepsilon)\cap S)\geq\delta\mu(B(x,\varepsilon)),\quad
0<\varepsilon\leq\lambda.$ (2)

This more general definition of standardness makes more sense in set
estimation, where it is typically assumed that $S$ is the support of a
probability distribution $\nu$. In that case, we say that $S$ is standard with
respect to $\nu$, or alternatively, that $\nu$ is standard. Note that, when
$\nu$ is the uniform distribution, standardness reduces to a shape constraint
on the set $S$, since it is satisfied whenever the set is standard with
respect to the Lebesgue measure. When the distribution is not uniform,
standardness represents an interplay between the geometry of the set and the
measure $\nu$, more than just a shape restriction on $S$. Therefore,
standardness becomes an additional regularity constraint to the possible shape
constraints of the set to be imposed. This condition was first introduced in
the context of set estimation in Cuevas (1990). It also appears in other
contexts under different names. For example, in the study of function spaces
(see the definitions of $d$-measure and $d$-set in Jonsson and Wallin (1984)).
In Benjamini et al. (2004) this $d$-set restriction is imposed to study the
“Hausdorff dimensions of various random sets associated to the Reflected
Brownian Motion”. Revisiting its application in set estimation, the
standardness condition can be found in Cuevas and Fraiman (1997); Cuevas and
Rodríguez-Casal (2004); Cholaquidis et al. (2014); Aaron and Bodart (2016);
Aaron et al. (2017), among many others. Proposition 1 in Aaron et al. (2017)
proves that, the (inside) $r$-rolling condition implies standardness with
respect to measures $\nu$ whose density with respect to the Lebesgue measure
is bounded from below by a positive constant. A double standardness assumption
was considered in Cuevas et al. (2013), for the problem of the estimation of
the Minkowski content. A generalization of the standardness assumption to
lower-dimensional sets can be found in Rinaldo and Wasserman (2010).

###  2.3 Standardness in set estimation

In what follows, $\aleph_{n}=\\{X_{1},\dots,X_{n}\\}$ will denote a sample of
independent observations generated from a distribution $\nu$ supported on a
set $S\subset\mathbb{R}^{d}$.

This is the most common sampling scheme in set estimation, and the one we will
consider in this work. There are alternatives to this i.i.d. model, such as
the Poisson model or the so-called inner-outer model, see Cuevas and Fraiman
(2009). In the Poisson model, the sample is given by observations of a Poisson
process whose intensity function is supported on $S$. In the inner-outer
model, the data consists of i.i.d. points, taken in a set containing $S$, and
for each observation, we are able to identify whether it belongs to $S$ or
not. A data-dependent sampling scheme, also considered in the literature,
assumes that the data comes from the trajectory of a stochastic process living
on $S$, observed in an interval $[0,T]$. This approach is of paramount
importance in ecology, where $S$ can represent the home-range of an animal,
and the goal is to estimate $S$ from a set of locations collected over a
period of time, see Cholaquidis et al. (2016, 2021); Baíllo and Chacón (2021)
and references therein.

Standardness arises, within the context of set estimation, in the study of the
convergence rate of the Hausdorff distance between $S$ and $\aleph_{n}$. To be
more specific, the hypothesis that $S$ is standard with respect to $\nu$ is
used in Cuevas and Rodríguez-Casal (2004) (Theorem 3) to demonstrate that, for
any $\epsilon>0$,

$\mathbb{P}(d_{H}(\aleph_{n},S)>2\epsilon)\leq
C\epsilon^{-d}\exp(-n\delta\omega_{d}\epsilon^{d}),$

where $C$ is a constant depending on $S$. This result, in turn, allows us to
obtain convergence rates for $d_{H}(\aleph_{n},S)$. As expected, the smaller
the value of $\delta$ (which can be interpreted in the uniform case as the set
$S$ having sharper peaks), the slower the rate. The standardness condition is
also relevant for obtaining convergence rates for the well-known Devroye-Wise
estimator, introduced in Devroye and Wise (1980), as well as for other usual
set estimators.

Given a sequence of positive real numbers, $\varepsilon_{n}\to 0$, the
Devroye-Wise estimator is given by

$\hat{S}_{n}=\bigcup_{i=1}^{n}B(X_{i},\varepsilon_{n}).$

The parameter $\varepsilon_{n}$ must be chosen by the practitioner and depends
on the standardness of $S$ with respect to $\nu$ (according to Cuevas and
Rodríguez-Casal (2004), it should be larger than
$[2\log(n)/(n\delta\omega_{d})]^{1/d}$, where $\delta$ is any value such that
(2) holds). Then, a proper choice for the constant $\delta$ is crucial, since
an underestimation produces a slower convergence. Moreover, several functional
of $S$ can be estimated by means of $\hat{S}_{n}$ and, therefore, the
convergence rates will also depend on a proper choice of $\delta$. For
instance, $\partial S$ can be consistently estimated by $\partial\hat{S}_{n}$,
choosing the value of $\varepsilon_{n}$ as before. The estimator $\hat{S}_{n}$
is also used as an input to define an estimator of the $(d-1)$-Lebesgue
measure of $\partial S$, see Aaron et al. (2022). Standardness plays an
important role also to determine, from a sample, if the support of the
distribution is lower dimensional or not (see Aaron et al. (2017) point (4) in
Theorem 1).

###  2.4 Topological data analysis

Standardness has also gained attention recently in persistent homology, where
a generalization, called $(a,b)$-standard assumption, is defined, see Chazal
et al. (2015b, a, 2016) It is said that a probability measure $\nu$ in
$\mathbb{R}^{d}$, whose support is a compact set $S$, satisfies the
$(a,b)$-standard assumption if there exist $a,b>0$, such that, for every $x\in
S$ and every $\varepsilon>0$, $\nu(B(x,\varepsilon))\geq 1\wedge
a\varepsilon^{b}$ (in the so-called $(a,b,r_{0})$-standard assumption, this
inequality holds for all $0\leq\varepsilon<r_{0}$). Note that the
$(a,b)$-standard assumption generalizes condition (2), since it allows for
$b>0$ to not necessarily be equal to $d$.

Let $\mathcal{P}$ denote a certain well-defined persistence diagram associated
with the support $S$ of a probability distribution $\nu$. In Fasy et al.
(2014), the authors obtain confidence sets for $\mathcal{P}$ based on an
estimate $\hat{\mathcal{P}}$, constructed from a sample $\aleph_{n}$ generated
from $\nu$. In the related paper Chazal et al. (2015b), convergence rates of
$\hat{\mathcal{P}}$ to $\mathcal{P}$ are obtained.

The connection between TDA and set estimation arises from the fundamental
property of stability in persistence diagrams Chazal et al. (2014). This
property allows us to demonstrate that the bottleneck distance (a usual metric
on the space of persistence diagrams) between ${\mathcal{P}}$ and
$\hat{\mathcal{P}}$, is upper bounded by $d_{H}(\aleph_{n},S)$. Therefore, the
problem of estimating persistence diagrams can be related to the problem of
estimating the support of a measure, with the standardness condition playing a
crucial role, as previously mentioned.

##  3 The standardness constant

If a set $S\subset\mathbb{R}^{d}$ is standard with respect to a Borel measure
$\nu$, as given in Definition 2.2, and (2) is satisfied for some pair
$(\delta,\lambda)$, then (2) also holds for any $\delta^{\prime}<\delta$ with
the same $\lambda$. Thus, we will define the standardness constant associated
to $S$ as the largest value for $\delta$, for which (2) holds.

######  Definition 3.1.

Assume that $S\subset\mathbb{R}^{d}$ is standard with respect to $\nu$ and let
$A$ be the set of all pairs
$(\delta,\lambda)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$ fulfilling (2). We
define the standardness constant

$\Upsilon(S,\nu)=\sup_{(\delta,\lambda)\in A}\delta.$ (3)

The following shape restriction (slightly more restrictive than standardness,
see Remark 1) will be required:

(H1) The support $S\subset\mathbb{R}^{d}$ of a probability measure $\nu$
satisfies

$0<\lim_{r\to 0}\inf_{x\in
S}\frac{\nu(B(x,r))}{r^{d}\omega_{d}}=:\delta_{\text{est}}<\infty.$ (4)

In case that the limit in (4) exists and is finite, the following Lemma states
that $S$ is standard with respect to $\nu$, with standardness constant
$\delta_{\text{est}}$.

######  Lemma 3.2.

Assume that the support $S\subset\mathbb{R}^{d}$ of a probability measure
$\nu$ fulfills condition (H1). Then, $S$ is standard with respect to $\nu$ and

$\Upsilon(S,\nu)=\lim_{r\to 0}\inf_{x\in
S}\frac{\nu(B(x,r))}{r^{d}\omega_{d}},$

where $\Upsilon(S,\nu)$ is the standardness constant defined by (3).

######  Remark 1.

The converse implication in Lemma 3.2 is not true in general, that is, there
exists a set $S$ and a measure $\nu$ such that $S$ is standard with respect to
$\nu$ but the limit (4) does not exists: consider $S\subset[0,1]$ the ternary
Cantor’s set and $\nu$ the measure defined by the Cantor’s function,
restricted to $S$. Standardness follows from Proposition 5.5 in Dovgoshey et
al. (2006), while the non-existence of the limit in (4) follows from
Proposition 9.1 in Dovgoshey et al. (2006).

Next, Proposition 1 states that if $\nu$ has a continuous density and $S$ is
standard with respect to $\mu$, then $\Upsilon(S,\nu)$ is either the minimum
of the density, or it is determined by an interplay between the density on the
boundary and the standardness of $S$ with respect to $\mu$. Lemma 3.3 states
if $S$ is compact, then the infimum in (4) is attained. The proof is left as
an easy exercise.

######  Lemma 3.3.

Let $S\subset\mathbb{R}^{d}$ and $\nu$ a probability distribution supported on
$S$ such that $\nu(\partial B(x,r))=0$ for all $r>0$ and $x\in S$. Then,
$\nu(B(x,r))$ is a continuous function of $(x,r)$.

######  Proposition 1.

Let $S\subset\mathbb{R}^{d}$ be a compact set such that

$0<\mathcal{L}_{\partial S}=\lim_{r\to 0}\min_{x\in\partial
S}\frac{\mu(B(x,r)\cap S)}{r^{d}\omega_{d}}<\infty.$ (5)

Assume that $\nu\ll\mu$ and the density $f$ of $\nu$ with respect to $\mu$ is
continuous and bounded from below on $S$. Then, $S$ is standard with respect
to $\nu$, and condition (H1) is fulfilled. Moreover,

$\Upsilon(S,\nu)=\min\Bigg{\\{}\lim_{r\to 0}\min_{x\in\partial
S}f(x)\frac{\mu(B(x,r)\cap S)}{r^{d}\omega_{d}},\min_{x\in S}f(x)\Bigg{\\}}.$
(6)

The following corollaries are a consequence of Proposition 1. Corollary 1,
states that, roughly speaking, when $\nu$ is uniformly distributed on $S$, the
value of $\Upsilon(S,\nu)$ is determined by the sharpness of the outward
peaks. Corollary 2, states that if the boundary is smooth, $\Upsilon(S,\nu)$
equals $1/(2\mu(S))$.

######  Corollary 1.

Let $S\subset\mathbb{R}^{d}$ be a compact set such that the limit (5) exists.
Then $S$ is standard with respect to $\mu/\mu(S)$ and

$\Upsilon(S,\mu/\mu(S))=\lim_{r\to 0}\min_{x\in\partial S}\frac{\mu(B(x,r)\cap
S)}{r^{d}\omega_{d}\mu(S)}.$

######  Corollary 2.

Let $S\subset\mathbb{R}^{d}$ such that $S$ fulfills the inside $r$-rolling
ball condition for some $r>0$. Then,

$\frac{\mathcal{L}_{\partial
S}}{\mu(S)}=\Upsilon(S,\mu/\mu(S))\geq\frac{1}{2\mu(S)},$

where $\mathcal{L}_{\partial S}$, is given by (5). Moreover, if $\partial S$
fulfills also the outside $r^{\prime}$-rolling ball condition, for some
$r^{\prime}>0$, $\mathcal{L}_{\partial
S}/\mu(S)=\Upsilon(S,\mu/\mu(S))=1/(2\mu(S)).$

##  4 Estimation of the standardness constant

Let $\aleph_{n}=\\{X_{1},\dots,X_{n}\\}$ be an i.i.d. sample of a random
vector $X$ whose distribution, $\nu$, is supported on a compact set
$S\subset\mathbb{R}^{d}$. First we prove that, under very general conditions,
the plug-in estimator

$\hat{\Upsilon}_{n}\equiv\hat{\Upsilon}_{n}(S,\nu)=\min_{j}\frac{\\#\\{\aleph_{n}\cap
B(X_{j},r_{n})\\}/n}{r_{n}^{d}\omega_{d}}.$ (7)

is a.s. consistent. Recall that the consistency in probability of (7) is
obtained in Fasy et al. (2014), under the hypotheses that $\nu$ has a density
with respect to Lebesgue measure, along with some other geometric restrictions
on $S$.

######  Theorem 4.1.

Let $\aleph_{n}=\\{X_{1},\dots,X_{n}\\}$ be an i.i.d. sample of a random
vector $X$ whose distribution, $\nu$, is supported on a compact set
$S\subset\mathbb{R}^{d}$, such that $\nu(\partial B(x,r))=0$ for all $r<r_{0}$
and $x\in S$. Assume that $S$ fulfills condition (H1). Denote by

$\Omega(r)=\Bigg{|}\min_{x\in S}\frac{\nu(B(x,r)\cap
S)}{\mu(B(x,r))}-\Upsilon(S,\nu)\Bigg{|}.$

Let $r_{n}$ be a sequence of positive real numbers such that $r_{n}\to 0$, and
$\hat{\Upsilon}_{n}$ be the estimator defined in (7). Then,

$\beta_{n}|\hat{\Upsilon}_{n}-\Upsilon(S,\nu)|\to 0\quad\text{a.s.},$ (8)

where $\beta_{n}\to\infty$ is such that $\beta_{n}\Omega(r_{n})\to 0$, and
$(r_{n}^{2d}/\beta_{n}^{2})(n/\log(n))^{1/d}\to\infty$.

######  Remark 2.

As we will see in the simulations (Section 5), in some cases the performance
of the plug-in estimator $\hat{\Upsilon}_{n}$ is quite poor. This happens
typically when $\min_{x\in S}\nu(B(x,r))/\mu(B(x,r))$ is attained at “too many
$x$”. This is the case, for instance, when
$S=B(0,1/\sqrt{\pi})\subset\mathbb{R}^{2}$ and $\nu$ is the uniform
distribution on $S$. For $n=3000,5000,9000$ the mean value of
$\hat{\Upsilon}_{n}$ over 500 replications was, respectively,
$0.4048,0.4131,0.4231$, showing that, even for very large values of $n$,
$\hat{\Upsilon}_{n}$ underestimates the true value $\Upsilon(S,\nu)=1/2$ (see
Table 1 for more results). Observe that in this case $\\{x\in
S:\Upsilon(S,\nu)=\lim_{r\to 0}\nu(B(x,r))/\mu(B(x,r))\\}=\partial S$ which
has $\nu$ measure 0. So, to ask

$\nu\left(\left\\{x\in S:\Upsilon(S,\nu)=\lim_{r\to
0}\frac{\nu(B(x,r))}{\mu(B(x,r))}\right\\}\right)=0,$ (9)

does not guarantee a good performance for $\hat{\Upsilon}_{n}$.

To address this underestimation, we suggest a correction for
$\hat{\Upsilon}_{n}$ in the following subsection which, roughly speaking,
takes into account the number of sample points $X_{i}$ at which the minimum
$\nu(B(X_{i},r_{n}))/\mu(B(X_{i},r_{n}))$ is attained.

###  4.1 A bias-correction method

Let us consider the estimator $\hat{\Upsilon}_{n}$ introduced in (7). Let
$r_{n}=\left(\log(n)/n\right)^{1/(2d)}$. We consider the correction:

$\tilde{\Upsilon}_{n}\equiv\tilde{\Upsilon}_{n}(S,\nu)=\hat{\Upsilon}_{n}\left(1+\frac{{\\#}\mathcal{A}_{n}}{n}\right),$

where

$\mathcal{A}_{n}=\left\\{i:\frac{\\#(\aleph_{n}\cap
B(X_{i},r_{n}))}{n\omega_{d}r_{n}^{d}}\leq\hat{\Upsilon}_{n}\Bigg{(}1+\omega_{d}r_{n}^{d/2}\Bigg{)}\right\\}.$

Then,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\mathcal{A}_{n}/n$
estimates the proportion of sample points $X_{i}$ such that
$\\#(\aleph_{n}\cap B(X_{i},r_{n}))/(n\omega_{d}r_{n}^{d})$ is “close” to the
minimum $\hat{\Upsilon}_{n}$.

######  Theorem 4.2.

Under the hypotheses of Theorem 4.1, and assuming also that (9) is fulfilled,

$\tilde{\Upsilon}_{n}\to\Upsilon(S,\nu)\quad a.s.$

##  5 Simulations

We conducted a small simulation study to show and compare the behavior of
estimators $\hat{\Upsilon}_{n}(S,\nu)$ and $\tilde{\Upsilon}_{n}(S,\nu)$. We
considered different situations by changing both the support $S$ and the
dimension of the space, as well as the probability distribution $\nu$.
Overall, the results confirm what was previously mentioned. The plug-in
estimator generally does not perform well, especially in cases where when
$\min_{x\in S}\nu(B(x,r))/\mu(B(x,r))$ is attained at many points. In those
cases, the estimator $\tilde{\Upsilon}_{n}(S,\nu)$ outperforms considerably
$\hat{\Upsilon}_{n}(S,\nu)$.

###  5.1 Uniform samples on $\mathbb{R}^{2}$

The objective here is to compare the performance of
$\hat{\Upsilon}_{n}(S,\nu)$ and $\tilde{\Upsilon}_{n}(S,\nu)$ when the sample
is uniformly distributed. To achieve this, we considered various sets in
$\mathbb{R}^{2}$, all of them with area one. Note that, with $\nu$ being the
uniform distribution and $S$ having area one, it follows that
$\Upsilon(S,\nu)\equiv\Upsilon(S,\mu)$. We examined the following sets:
$S_{1}$ is an equilateral triangle, $S_{2}=[0,1]^{2}$ is the unit square,
$S_{3}$ is a regular hexagon, and $S_{4}=B(0,1/\sqrt{\pi})$. It follows that
$\Upsilon(S_{1},\mu)=1/6$, $\Upsilon(S_{2},\mu)=1/4$,
$\Upsilon(S_{3},\mu)=1/3$ and $\Upsilon(S_{4},\mu)=1/2$.

Mean values of $\hat{\Upsilon}_{n}(S_{i},\nu)$ and
$\tilde{\Upsilon}_{n}(S_{i},\nu)$ over 500 replications are shown in Table 1,
for $i=1,\ldots,4$. In brackets we show the variance of the estimators. In all
cases $r_{n}=(\log(n)/n)^{1/4}$.

| $n=1000$ | $n=3000$ | $n=5000$ | $n=7000$ | $n=9000$
---|---|---|---|---|---
$\hat{\Upsilon}_{n}(S_{1},\mu)$ | 0.1765 | 0.1758 | 0.1745 | 0.1730 | 0.1735
| (0.0005) | (0.0003) | (0.0002) | (0.0002) | (0.0001)
$\tilde{\Upsilon}_{n}(S_{1},\mu)$ | 0.1829 | 0.1782 | 0.1761 | 0.1742 | 0.1745
| (0.0007) | (0.0004) | (0.0002) | (0.0002) | (0.0001)
$\hat{\Upsilon}_{n}(S_{2},\mu)$ | 0.2510 | 0.2520 | 0.2527 | 0.2523 | 0.2521
| (0.0006) | (0.0003) | (0.0003) | (0.0002) | (0.0002)
$\tilde{\Upsilon}_{n}(S_{2},\mu)$ | 0.2767 | 0.2606 | 0.2581 | 0.2563 | 0.2552
| (0.0014) | (0.0004) | (0.0003) | (0.0002) | (0.0002)
$\hat{\Upsilon}_{n}(S_{3},\mu)$ | 0.3190 | 0.3236 | 0.3246 | 0.3251 | 0.3250
| (0.0005) | (0.0003) | (0.0002) | (0.0002) | (0.0002)
$\tilde{\Upsilon}_{n}(S_{3},\mu)$ | 0.4014 | 0.3650 | 0.3519 | 0.3453 | 0.3407
| (0.0023) | (0.0009) | (0.0006) | (0.0004) | (0.0003)
$\hat{\Upsilon}_{n}(S,\mu)$ | 0.3813 | 0.4048 | 0.4131 | 0.4201 | 0.4231
| (0.0004) | (0.0002) | (0.0002) | (0.0001) | (0.0001)
$\tilde{\Upsilon}_{n}(S_{4},\mu)$ | 0.5380 | 0.5190 | 0.5088 | 0.5064 | 0.5015
| (0.0025) | (0.0008) | (0.0006) | (0.0005) | (0.0004)
Table 1: Mean values of $\hat{\Upsilon}_{n}(S_{i},\mu)$ and
$\tilde{\Upsilon}_{n}(S_{i},\mu)$ over 500 replications (variance shown in
brackets). Samples are generated from the uniform distribution.

###  5.2 Uniform samples on $\mathbb{R}^{d}$, for $d>2$

Table 1 illustrates that the estimator $\tilde{\Upsilon}_{n}$ significantly
outperforms $\hat{\Upsilon}_{n}$. This performance gap increases with the
dimension, as we will show here. We consider the uniform distribution $\nu$ on
$S=B(0,R)\subset\mathbb{R}^{d}$, where $R$ is chosen such that $\mu(S)=1$, for
$d=3$ and $d=4$. In both cases, $\Upsilon(S,\nu)\equiv\Upsilon(S,\mu)=1/2$. As
before, we replicated the whole procedure 500 times and report on Table 2 the
mean values of $\hat{\Upsilon}_{n}(S,\mu)$ and $\tilde{\Upsilon}_{n}(S,\mu)$
(variance between brackets). We have used $r_{n}=(\log(n)/n)^{1/(2d)}$.

| $n=1000$ | $n=3000$ | $n=5000$ | $n=7000$ | $n=9000$
---|---|---|---|---|---
| $d=3$
$\hat{\Upsilon}_{n}(S,\mu)$ | 0.3110 | 0.3385 | 0.3496 | 0.3579 | 0.3634
| (0.00021) | (0.00011) | (0.00009) | (0.00008) | (0.00007)
$\tilde{\Upsilon}_{n}(S,\mu)$ | 0.5209 | 0.5203 | 0.5151 | 0.5139 | 0.5119
| (0.00150) | (0.00072) | (0.00055) | (0.00044) | (0.00040)
| $d=4$
$\hat{\Upsilon}_{n}(S,\mu)$ | 0.2485 | 0.2766 | 0.2897 | 0.2982 | 0.3043
| (0.00013) | (0.00007) | (0.00006) | (0.00005) | (0.00004)
$\tilde{\Upsilon}_{n}(S,\mu)$ | 0.4438 | 0.4620 | 0.4677 | 0.4712 | 0.4728
| (0.00087) | (0.00047) | (0.00037) | (0.00033) | (0.00026)
Table 2: Mean values of $\hat{\Upsilon}_{n}(S,\mu)$ and
$\tilde{\Upsilon}_{n}(S,\mu)$ over 500 replications (variance shown in
brackets). For each value of $d$, the set $S$ is a $d$-ball
$S=B(0,R)\subset\mathbb{R}^{d}$ with $\mu(S)=1$. Samples are generated from
the uniform distribution on $S$.

###  5.3 Non-uniform samples

For the non-uniform case, we consider $\nu$ to be a mixture of distributions.
Specifically, let $\theta_{1}$ and $\theta_{2}$ be uniform distributions on
the interval $[0,2\pi]$, and let $R_{1}$ and $R_{2}$ be uniform distributions
on the interval $[0,1]$. We define
$Y_{1}=R_{1}^{1/2}\left(\cos(\theta_{1}),\sin(\theta_{1})\right)$, which is
uniformly distributed in the unit ball and
$Y_{2}=R_{2}^{1/4}\left(\cos(\theta_{2}),\sin(\theta_{2})\right)$, whose
distribution has an “antimode” at the origin $(0,0)$. We consider the mixture
$Y=\frac{1}{4}(Y_{1}+3Y_{2})$, with support $S=B(0,1)$.

In this case the standardness constant is $\Upsilon(S,\nu)=1/(4\pi)\approx
0.07958$. Mean values of $\hat{\Upsilon}_{n}(S,\nu)$ and
$\tilde{\Upsilon}_{n}(S,\nu)$ over 500 replications are shown in Table 3. In
brackets we show the variance of the estimators. In all cases
$r_{n}=(\log(n)/n)^{1/4}$.

| $n=1000$ | $n=3000$ | $n=5000$ | $n=7000$ | $n=9000$
---|---|---|---|---|---
$\hat{\Upsilon}_{n}(S,\mu)$ | 0.0857 | 0.0789 | 0.0775 | 0.0768 | 0.0771
| (0.000212) | (0.000109) | (0.000073) | (0.000064) | (0.000060)
$\tilde{\Upsilon}_{n}(S,\mu)$ | 0.0924 | 0.0817 | 0.0798 | 0.0787 | 0.0789
| (0.000399) | (0.000145) | (0.000093) | (0.000081) | (0.000074)
Table 3: Mean values of $\hat{\Upsilon}_{n}(S,\mu)$ and
$\tilde{\Upsilon}_{n}(S,\mu)$ over 500 replications (variance shown in
brackets). Samples are generated from a non-uniform distribution on
$S=B(0,1)\subset\mathbb{R}^{2}$.

##  6 Estimability

Our theoretical analysis, presented in Section 4, demonstrates that the
standardness constant can be estimated consistently. Nevertheless, we prove
here that determining the standardness of a set from a sample is not possible.
As previously mentioned in the introduction, the problem of consistently
estimating functionals of a random variable has a long-standing history dating
back to the pioneering works of Bahadur and Savage Bahadur and Savage (1956)
and Le Cam and Schwartz LeCam and Schwartz (1960). In this paper we contribute
to this ongoing discussion by approaching the non-estimability problem from a
set-estimation perspective. More precisely, if $\mathcal{M}$ denotes the
separable metric space of Borel probability distributions on $\mathbb{R}^{d}$
endowed with the total variation norm $\|\cdot\|_{TV}$, we aim to prove that
the functionals

$\alpha(\nu)=\begin{cases}1&\quad\text{ if }\nu\ll\mu,\\\ 0&\quad\text{
otherwise,}\end{cases}$ (10) $\beta(\nu)=\begin{cases}1&\quad\text{ if
}{\textnormal{supp}}(\nu)=\overline{{\textnormal{int}}({\textnormal{supp}}(\nu)),}\\\
0&\quad\text{ otherwise}\end{cases}$ (11)

and

$\gamma(\nu)=\begin{cases}1&\quad\text{ if }{\textnormal{supp}}(\nu)\text{ is
standard with respect to }\ \nu,\\\ 0&\quad\text{ otherwise,}\end{cases}$ (12)

cannot be consistently estimated from a sequence of i.i.d. observations. To
this aim we prove the following Lemma, whose proof is similar to that of Lemma
1.1 in Fraiman and Meloche (1999).

######  Lemma 6.1.

Let $\mathcal{B}\subset\mathcal{M}$ be equipped with some norm $\|\cdot\|$
such that $(\mathcal{B},\|\cdot\|)$ is a complete metric space. Assume that
$\|\nu\|_{TV}\leq c\|\cdot\|$ for some constant $c$ and all
$\nu\in\mathcal{B}$. Let $\Phi:\mathcal{B}\to[-K,K]$ be any bounded
characteristic of the distributions in $\mathcal{B}$. If $\phi$ is
consistently estimable on $\mathcal{B}$, then there exists a dense subset of
points in $\mathcal{B}$ at which $\phi$ is continuous with respect to the
topology induced by $\|\cdot\|$. Consequently, if $\phi$ is discontinuous at
every point in $\mathcal{B}$, then it is not consistently estimable on
$\mathcal{B}$.

######  Theorem 6.2.

Let $\mathcal{M}$ be a separable metric space of Borel probability
distributions on $\mathbb{R}^{d}$ endowed with the total variation norm. The
functional $\alpha$ defined by (10) cannot be consistently estimated from a
sequence of i.i.d. observations.

######  Theorem 6.3.

Let $\mathcal{M}$ be the class of all probability distributions on
$\mathbb{R}^{d}$ endowed with the total variation norm. The functionals,
defined on $\mathcal{M}$ by (11) and (12) cannot be consistently estimated
from a sequence of i.i.d. observations.

## Appendix A

In this section we provide the proofs of the theoretical results presented in
Sections 3 and 4 .

###### Proof of Lemma 3.2.

Let us denote $\delta_{\text{est}}=\lim_{\varepsilon\rightarrow 0}\inf_{x\in
S}\psi(x,\varepsilon)$, where

$\psi(x,\varepsilon)=\nu(B(x,\varepsilon)\cap S)/(\varepsilon^{d}\omega_{d}).$

Let $A$ be the set of all pairs $(\delta,\lambda)$ fulfilling (2), and
$\Upsilon(S,\nu)=\sup_{(\delta,\lambda)\in A}\delta$. From H1 it follows that,
from $r$ small enough, $A$ is non-empty. We have to prove that
$\delta_{\text{est}}=\Upsilon(S,\nu)$. Let us start by proving that
$\Upsilon(S,\nu)\leq\delta_{\text{est}}$. Let us fix $(\delta,\lambda)\in A$.
Then $\psi(x,\varepsilon)\geq\delta$ for all $x\in S$ and
$0<\varepsilon<\lambda$. Then, we have

$\delta_{\text{est}}=\lim_{\varepsilon\rightarrow 0}\inf_{x\in
S}\psi(x,\varepsilon)\geq\delta.$

Therefore, it follows that $\Upsilon(S,\nu)\leq\delta_{\text{est}}$, because
$\Upsilon(S,\nu)$ is the smallest upper bound.

To prove $\Upsilon(S,\nu)\geq\delta_{\text{est}}$, let us assume that
$\Upsilon(S,\nu)<\delta_{\text{est}}$. Define
$\gamma^{*}=(\delta_{\text{est}}-\Upsilon(S,\nu))/2$. There exists
$\varepsilon^{*}>0$ such that, for all $0<\varepsilon<\varepsilon^{*}$,

$\inf_{x\in S}\psi(x,\varepsilon)\in
B\left(\delta_{\text{est}},\gamma^{*}\right).$

Then, for all $x\in S$ and $0<\varepsilon<\varepsilon^{*}$,

$\Upsilon(S,\nu)<\delta^{*}<\inf_{x\in
S}\psi(x,\varepsilon)\leq\psi(x,\varepsilon),$

where $\delta^{*}=\delta_{\text{est}}-\gamma^{*}$. This implies that
$(\delta^{*},\varepsilon^{*})\in A$ and $\delta^{*}>\Upsilon(S,\nu)$ which is
a contradiction. ∎

###### Proof of Proposition 1.

Let us first prove that (5) implies that $S$ is standard with respect to
$\mu$: let $0<\varepsilon<\mathcal{L}_{\partial S}/4$, take $r_{0}$ such that
$|\mathcal{L}_{\partial S}-\min_{x\in\partial S}\mu(B(x,r)\cap
S)/(r^{d}\omega_{d})|<\varepsilon$, for all $r<r_{0}$. For all $x\in S$ such
that $d(x,\partial S)\geq r_{0}$,

$\frac{\mu(B(x,2r)\cap
S)}{(2r)^{d}\omega_{d}}\geq\frac{1}{2^{d}}\frac{\mu(B(x,r)\cap
S)}{r^{d}\omega_{d}}=\frac{1}{2^{d}}.$

If $d(x,\partial S)\leq r_{0}$, let $y$ the closest point to $x$ in $\partial
S$, then

$\frac{\mu(B(x,2r)\cap
S)}{(2r)^{d}\omega_{d}}\geq\frac{1}{2^{d}}\frac{\mu(B(y,r)\cap
S)}{r^{d}\omega_{d}}=\frac{1}{2^{d}}\min_{y\in\partial S}\frac{\mu(B(y,r)\cap
S)}{r^{d}\omega_{d}}\geq\frac{3\mathcal{L}_{\partial S}}{2^{d+2}}.$

Since $f$ is bounded from below by a positive constant, and $S$ is standard
with respect to $\mu$, it follows easily that $S$ is standard with respect to
$\nu$.

Let us prove that $S$ fulfills condition (H1). From Lemma 3.2, we have to
prove that $\lim_{r\to 0}\min_{x\in S}\psi(x,r)$ exists and is given by (6).
Let $x_{0}\in\text{argmin}_{x\in S}f(x)$. Since $S$ is compact and $f$ is
continuous, it is uniformly continuous, that is, for all $\varepsilon>0$ there
exists $\delta>0$ such that $|f(x)-f(y)|\leq\varepsilon$ for all
$\|x-y\|<\delta$. Let $\varepsilon>0$ and $r<\delta/2$. Then,

$\psi(x_{0},r):=\frac{\int_{B(x_{0},r)\cap
S}f(t)dt}{r^{d}\omega_{d}}\leq(f(x_{0})+\varepsilon)\frac{\mu(B(x_{0},r)\cap
S)}{r^{d}\omega_{d}}\leq f(x_{0})+\varepsilon.$

Then, it follows that $\limsup_{r\to 0}\min_{x\in S}\psi(x,r)\leq\min_{t\in
S}f(t)$. Let $r<\delta/2$,

$\min_{x\in S}\psi(x,r)\leq\min_{x\in\partial S}\frac{\int_{B(x,r)\cap
S}f(t)dt}{r^{d}\omega_{d}}\leq\varepsilon+\min_{x\in\partial
S}f(x)\frac{\mu(B(x,r)\cap S)}{r^{d}\omega_{d}}.$

Then,

$\limsup_{r\to 0}\min_{x\in
S}\psi(x,r)\leq\min\Bigg{\\{}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\min_{x\in
S}f(x)},\lim_{r\to 0}\min_{x\in\partial S}f(x)\frac{\mu(B(x,r)\cap
S)}{r^{d}\omega_{d}}\Bigg{\\}}.$

It remains to be proved that $\liminf_{r\to 0}\min_{x\in S}\psi(x,r)$ is
bounded from below by the right-hand side of (6). Since $\psi(x,r)$ is a
continuous function of $x$ we can choose $r_{n}\to 0$ and $x_{n}$ such that

$\psi(x_{n},r_{n})=\min_{x\in S}\psi(x,r_{n})\quad\text{ and
}\quad\psi(x_{n},r_{n})\to\liminf_{r\to 0}\min_{x\in S}\psi(x,r).$

We can assume, by considering a subsequence if necessary, that $x_{n}\to
x^{*}\in S$. If $x^{*}\in{\textnormal{int}}(S)$, then there exists $\eta>0$
such that $B(x^{*},\eta)\subset S$. Let $\eta^{*}=\min\\{\eta/2,\delta/2\\}$,
being $\delta$ from the uniform continuity of $f$. Then, for all $n$ large
enough such that $r_{n}<\eta^{*}$, $\psi(x_{n},r_{n})>f(x^{*})+\varepsilon$,
from where it follows that $\liminf_{r\to 0}\min_{x\in S}\psi(x,r)\geq
f(x^{*})\geq\min_{x\in S}f(x)$. If $x^{*}\in\partial S$, for all $n$ large
enough such that $r_{n}<\eta^{*}$,

$\psi(x_{n},r_{n})\geq(f(x^{*})-\varepsilon)\frac{\mu(B(x^{*},r_{n})\cap
S)}{\mu(B(x^{*},r_{n}))}\geq f(x^{*})\frac{\mu(B(x^{*},r_{n})\cap
S)}{\mu(B(x^{*},r_{n}))}-\varepsilon\\\ \geq\min_{x\in\partial
S}f(x)\frac{\mu(B(x,r_{n})\cap S)}{\mu(B(x,r_{n}))}-\varepsilon,$

from where it follows that

$\liminf_{r\to 0}\min_{x\in S}\psi(x,r)\geq\lim_{r\to 0}\min_{x\in\partial
S}f(x)\frac{\mu(B(x,r)\cap S)}{r^{d}\omega_{d}},$

which concludes the proof. ∎

###### Proof of Theorem 4.1.

From Corollary 13.2 in Devroye et al. (2013), $\mathcal{C}=\\{B(x,r_{n}):x\in
S,n\in\mathbb{N}\\}$ has Vapnik–Chervonenkis dimension at most $d+2$. Then,
from Theorems 12.5 and 13.3 in Devroye et al. (2013),

$\mathbb{P}\left(\sup_{B=B(x,r_{n})\in\mathcal{C}}\left|\frac{\\#\\{\aleph_{n}\cap
B\\}}{n}-\nu(B)\right|>\varepsilon\frac{r_{n}^{d}}{\beta_{n}}\omega_{d}\right)\leq
8n^{d+2}\exp\Big{(}\frac{-n\varepsilon^{2}r_{n}^{2d}\omega_{d}^{2}}{32\beta_{n}^{2}}\Big{)}.$

Furthermore, from Borel Cantelli’s lemma,

$\beta_{n}\sup_{B=B(x,r_{n})\in\mathcal{C}}\frac{1}{r_{n}^{d}\omega_{d}}\left|\frac{\\#\\{\aleph_{n}\cap
B\\}}{n}-\nu(B)\right|\to 0\quad\text{ a.s}.$ (13)

Let $x_{r}$ denote a point in $S$ that minimizes
$\nu(B(x,r))/(r^{d}\omega_{d})$, and $X_{i,r_{n}}$ the closest point in
$\aleph_{n}$ to $x_{r_{n}}$.

Then, with probability one, for $n$ large enough,

$\|X_{i,r_{n}}-x_{r_{n}}\|\leq d_{H}(\aleph_{n},S)\leq K(\log(n)/n)^{1/d},$

for some $K>0$, (see Theorem 3 in Cuevas and Rodríguez-Casal (2004)). Let us
denote $\varepsilon_{n}=K(\log(n)/n)^{1/d}$. Observe that
$\varepsilon_{n}/r_{n}\to 0$ since by hypothesis
$(\varepsilon_{n}/r_{n})(\beta_{n}^{2}/r_{n}^{2d-1})\to 0$ and
$\beta_{n}^{2}/r_{n}^{2d-1}$ to $\infty$.

Then, with probability one, for $n$ large enough,

$\beta_{n}\Bigg{|}\frac{\nu(B(X_{i,r_{n}},r_{n}))}{r_{n}^{d}\omega_{d}}-\frac{\nu(B(x_{r_{n}},r_{n}))}{r_{n}^{d}\omega_{d}}\Bigg{|}\leq\beta_{n}\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{r_{n}^{d}\omega_{d}}-\beta_{n}\frac{\nu(B(x_{r_{n}},r_{n}))}{r_{n}^{d}\omega_{d}}\\\
\leq\beta_{n}\Bigg{|}\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{r_{n}^{d}\omega_{d}}-\Upsilon(S,\nu)\Bigg{|}+\beta_{n}\Omega(r_{n}).$
(14)

Since

$\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{r_{n}^{d}\omega_{d}}=\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{(r_{n}+\varepsilon_{n})^{d}\omega_{d}}\frac{(r_{n}+\varepsilon_{n})^{d}}{r_{n}^{d}},$

it follows that

$\Bigg{|}\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{r_{n}^{d}\omega_{d}}-\Upsilon(S,\nu)\Bigg{|}\leq\Omega(r_{n}+\varepsilon_{n})\Big{(}1+\frac{\varepsilon_{n}}{r_{n}}\Big{)}^{d}+\\\
\Upsilon(S,\nu)\Bigg{|}1-\Big{(}1+\frac{\varepsilon_{n}}{r_{n}}\Big{)}^{d}\Bigg{|}.$

If use that $(1+x)^{d}=1+dx+o(x^{2})$, then, for $n$ large enough,

$\Bigg{|}\frac{\nu(B(x_{r_{n}},r_{n}+\varepsilon_{n}))}{r_{n}^{d}\omega_{d}}-\Upsilon(S,\nu)\Bigg{|}\leq
2\Omega(r_{n}+\varepsilon_{n})+\Upsilon(S,\nu)d\frac{\varepsilon_{n}}{r_{n}}+o\Big{(}\frac{\varepsilon_{n}^{2}}{r_{n}^{2}}\Big{)}.$

Since $\beta_{n}\varepsilon_{n}/r_{n}\to 0$, and $\beta_{n}\Omega(r_{n})\to
0$,

$\beta_{n}\Bigg{|}\frac{\nu(B(X_{i,r_{n}},r_{n}))}{r_{n}^{d}\omega_{d}}-\frac{\nu(B(x_{r_{n}},r_{n}))}{r_{n}^{d}\omega_{d}}\Bigg{|}\to
0.$ (15)

From (15), (13), and $\beta_{n}\Omega(r_{n})\to 0$,

$\beta_{n}\Bigg{|}\frac{\\#\\{\aleph_{n}\cap
B(X_{i,r_{n}},r_{n})\\}/n}{r_{n}^{d}\omega_{d}}-\Upsilon(S,\nu)\Bigg{|}\to 0.$

Then

$\beta_{n}(\hat{\Upsilon}_{n}-\Upsilon(S,\nu))\leq\beta_{n}\Bigg{|}\frac{\\#\\{\aleph_{n}\cap
B(X_{i,r_{n}},r_{n})\\}/n}{r_{n}^{d}\omega_{d}}-\Upsilon(S,\nu)\Bigg{|}\to 0.$

This proves that, with probability one, for all $n$ such that
$\beta_{n}(\hat{\Upsilon}_{n}-\Upsilon(S,\nu))\geq 0$,
$\beta_{n}(\hat{\Upsilon}_{n}-\Upsilon(S,\nu))\to 0$ a.s. Let us define
$\mathcal{N}=\\{n\in\mathbb{N}:\beta_{n}(\hat{\Upsilon}_{n}-\Upsilon(S,\nu))<0\\}$.
If $\\#\mathcal{N}<\infty$, then (8) follows. Let us consider the case
$\\#\mathcal{N}=\infty$. Assume by contradiction that there exists $\gamma>0$
such that, with positive probability,

$\beta_{n}(\Upsilon(S,\nu)-\hat{\Upsilon}_{n})>\gamma\quad\text{ for all
}n>n_{0},n\in\mathcal{N}.$

For all $n\in\mathcal{N}$ and $n>n_{0}$ let $X_{j,r_{n}}$ a sample point that
minimize $\\#\\{\aleph_{n}\cap B(X_{i},r_{n})\\}/(r_{n}^{2}\omega_{d})$. From
(13), for $n>n_{1}\geq n_{0}$

$\beta_{n}\Bigg{(}\Upsilon(S,\nu)-\frac{\nu(B(X_{j,r_{n}},r_{n})\cap
S)}{r_{n}^{d}\omega_{d}}\Bigg{)}>\gamma/2>0.$

Then, with positive probability, for all $n\in\mathcal{N}$ with $n>n_{1}$,

$\beta_{n}\Bigg{(}\Upsilon(S,\nu)-\min_{x\in S}\frac{\nu(B(x,r_{n})\cap
S)}{r_{n}^{d}\omega_{d}}\Bigg{)}+\\\ \beta_{n}\Bigg{(}\min_{x\in
S}\frac{\nu(B(x,r_{n})\cap
S)}{r_{n}^{d}\omega_{d}}-\frac{\nu(B(X_{j,r_{n}},r_{n})\cap
S)}{r_{n}^{d}\omega_{d}}\Bigg{)}>\gamma/2>0,$ (16)

which is a contradiction because the second term is negative or zero, and the
first one converges to 0. ∎

###### Proof of Theorem 4.2.

To prove that $\tilde{\Upsilon}_{n}\to\hat{\Upsilon}_{n}$ it is enough to
prove that
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\mathcal{A}_{n}/n\to
0$.

Let $\beta_{n}$ as in Theorem 4.1. For all $\varepsilon>0$, with probability
one, for all $n$ large enough,
$\hat{\Upsilon}_{n}\leq\Upsilon(S,\nu)+\varepsilon/\beta_{n}$. Then there
exists $C$ a positive constant such that, with probability one, for all $n$
large enough,

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\mathcal{A}_{n}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\left\\{i:\frac{\\#(\aleph_{n}\cap
B(X_{i},r_{n}))}{n\omega_{d}r_{n}^{d}}\leq\Upsilon(S,\nu)+\frac{\varepsilon}{\beta_{n}}+Cr_{n}^{d/2}\right\\}:={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
From (13), with probability one, for all $n$ large enough,

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\mathcal{B}_{n}\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\left\\{i:\frac{\nu(B(X_{i},r_{n}))}{\omega_{d}r_{n}^{d}}\leq\Upsilon(S,\nu)+2\frac{\varepsilon}{\beta_{n}}+Cr_{n}^{d/2}\right\\}.$

Let denote

$\mathbb{C}_{n}=\left\\{x\in
S:\frac{\nu(B(x,r_{n}))}{\omega_{d}r_{n}^{d}}\leq\Upsilon(S,\nu)+2\frac{\varepsilon}{\beta_{n}}+Cr_{n}^{d/2}\right\\},$

and

$\mathbb{C}=\left\\{x\in S:\Upsilon(S,\nu)=\lim_{r\to
0}\frac{\nu(B(x,r))}{\omega_{d}r^{d}}\right\\},$

then, with probability one, for $n$ large enough,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\#}\mathcal{A}_{n}/n\leq(1/n)\sum_{i=1}^{n}\mathbb{I}_{\mathbb{C}_{n}}(X_{i})$.
From Hoeffding inequality’s it follows that

$\Bigg{|}\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}_{\mathbb{C}_{n}}(X_{i})-\nu(\mathbb{C}_{n})\Bigg{|}\to
0,\quad a.s.$

Observe that $\mathbb{I}_{\mathbb{C}_{n}}(y)=0$ for $n$ large enough, for all
$y\in\mathbb{C}^{c}$, from where it follows that $\nu(\mathbb{C}_{n})\to 0$,
which concludes the proof. ∎

## Appendix B

In this section we provide the proofs of the theoretical results presented in
Section 6.

###### Proof of Lemma 6.1.

Suppose that $\phi$ is estimable in $\mathcal{B}$, from $X_{1},\dots,X_{n}$
i.i.d. with distribution $\nu$, with the estimator $T_{n}(X_{1},\dots,X_{n})$.
Let us define

$S_{n}(X_{1},\dots,X_{n})=T_{n}\mathbb{I}_{|T_{n}|\leq
K}+\text{sign}(T_{n})K\mathbb{I}_{|T_{n}|>K}.$

As in Fraiman and Meloche (1999), $S_{n}$ is a consistent estimate of $\phi$
on $\mathbb{B}$. Define $\phi_{n}:\mathcal{B}\to[-K,K]$ by

$\displaystyle\phi_{n}(\nu)=$
$\displaystyle\mathbb{E}_{\nu}(S_{n})=\int\dots\int
S_{n}(x_{1},\dots,x_{n})d\nu\dots d\nu$ $\displaystyle=$
$\displaystyle\int_{(\mathbb{R}^{d})^{n}}S_{n}(x_{1},\dots,x_{n})d\nu^{n}(x_{1},\dots,x_{n}),$

where $\nu^{n}$ is the product measure of $\nu$, $n$ times. Since $S_{n}$ is
bounded $\phi_{n}(\nu)\to\phi(\nu)$, for all $\nu\in\mathcal{B}$. In addition,
for all $\nu,\mu\in\mathcal{B}$,

$|\phi_{n}(\nu)-\phi_{n}(\mu)|=|\mathbb{E}_{\nu}(S_{n})-\mathbb{E}_{\mu}(S_{n})|\leq
K\|\nu^{n}-\mu^{n}\|_{TV}\\\ \leq Kn\|\nu-\mu\|_{TV}\leq ncK\|\nu-\mu\|.$

Now as in Fraiman and Meloche (1999), using Baire’s theorem, it follows that
$\phi$ must be continuous on a dense subset of $\mathcal{B}$. ∎

###### Proof of Theorem 6.2.

From Lemma 6.1 it is enough to prove that $\alpha$ is discontinuous at every
$\nu\in\mathcal{M}$. Let us consider $\nu\in\mathcal{M}$ such that
$\nu\ll\mu$. For all $\varepsilon>0$ there exists $x$ and $r>0$ such that
$\nu(B(x,r))<\varepsilon/2$, where it can be $\nu(B(x,r))=0$. Let $z,y\in
B(x,r)$ such that $z\neq y$. Let $U$ be a random variable with uniform
distribution on the segment $[z,y]$ joining $z$ and $y$. Let us consider $B$ a
Bernoulli random variable with $\mathbb{P}(B=1)=\varepsilon/2$. Let us
consider a random vector $X$ such that if $B=1$ then $X$ is distributed as the
restriction of $\nu$ to ${\textnormal{supp}}(\nu)\setminus B(x,r)$, on the
contrary if $B=0$, $X=U$. Let $\lambda$ be the distribution of $X$, then
$\lambda$ is not absolutely continuous with respect to the Lebesgue measure
$\mu$, and $\|\lambda-\nu\|_{TV}<\varepsilon$.

Let us consider now $\nu$ such that $\alpha(\nu)=0$. Let $\lambda_{n}$ be the
probability measure on $\mathbb{R}^{d}$ associated to a Gaussian kernel
$K_{n}$, with mean $0$ and variance $(1/n)I_{d}$ being $I_{d}$ the identity
matrix in $\mathbb{R}^{d}$. Let $\nu\ast\lambda_{n}$ be the convolution of
$\nu$ and $\lambda_{n}$, then it has density $K_{n}\ast\nu$ defined by

$K_{n}\ast\nu(y)=\int K_{n}(y-x)d\nu(x).$ (17)

Since $\lambda_{n}\to\delta_{0}$ the point mass in 0, and
$\nu\ast\delta_{0}=\nu$, from Parthasarathy and Steerneman (1985),
$\lambda_{n}\ast\nu\to\nu$ in total variation, which concludes the proof of
the theorem. ∎

###### Proof of Theorem 6.3.

Let us start by proving that (11) cannot be consistently estimated. We will
apply Lemma 6.1. Let $\nu\in\mathcal{M}$ be such that $\beta(\nu)=0$. Let us
denote $S={\textnormal{supp}}(\nu)$. Let us consider the sequence of sets
$S_{n}=B(S,1/n)$. Then $\overline{{\textnormal{int}}(S_{n})}=S_{n}$. Since $S$
is closed, $\cap_{n}S_{n}=S$ . Let us consider $\lambda$ the standard Gaussian
measure on $\mathbb{R}^{d}$. For all $\varepsilon>0$ there exists $n$ large
enough such that $\lambda(S_{n}\setminus S)<\varepsilon$. The measure $\xi$
defined as

$\xi(A)=\frac{\nu(A\cap S)+\lambda(A\cap(S_{n}\setminus
S))}{1+\lambda(S_{n}\setminus S)},$

where $A$ is a Borel set, is a probability measure supported on $S_{n}$,
$\beta(\xi)=1$ and $\|\xi-\nu\|_{TV}<2\varepsilon$. Let $S$ such that
$\beta(S)=1$. We can proceed as in the proof of Theorem 6.2; let
$\varepsilon>0$, we first consider $B(x,r)$ such that
$\nu(B(x,r))<\varepsilon$ (it can be $\nu(B(x,r))=0$), then we consider a non-
empty segment on $B(x,r)$ define the uniform on that segment. The measure
$\lambda$ defined as in Theorem 6.2 fulfills that $\beta(\lambda)=0$ and
$\|\lambda-\nu\|_{TV}<\varepsilon$.

To prove that (12) cannot be consistently estimate let us consider first
$\nu\in\mathcal{M}$ such that $\gamma(\nu)=0$. Let $\varepsilon>0$ and $R$
large enough such that $\nu(B(0,R)^{c})<\varepsilon$. Let
$S_{\varepsilon}=S\cap B(0,R)$ and $S_{n}=B(S_{\varepsilon},1/n)$. Observe
that $S_{n}$ is standard with respect to the $\mu$. Reasoning as before,
$S_{\varepsilon}$ is closed and then $\cap_{n}S_{n}=S$. Let us consider
$\nu\ast\lambda_{n}$ the convolution of $\nu$ with $\lambda_{n}$, the Gaussian
measure on $\mathbb{R}^{d}$ with variance $1/n$, as in Lemma 6.1. There exists
$n$ large enough such that $\nu*\lambda_{n}(S_{n}\setminus
S_{\varepsilon})<\varepsilon$. The density of $\nu*\lambda_{n}$ is given by
(17). Let $M$ be a compact set such that $\nu(M)>0$, then for all $z\in S_{n}$
and all $r>0$,

$\nu*\lambda_{n}(B(z,r)\cap S_{n})=\int_{B(z,r)\cap
S_{n}}\int_{\mathbb{R}^{d}}K_{n}(y-x)d\nu(x)dy\geq\\\ \int_{B(z,r)\cap
S_{n}}K_{n}(y-x)d\nu(x)dy.$

If $z\in S_{n}$ and $y\in B(z,r)\cap S_{n}$ then $y\in B(S_{n},r)$ which is a
compact set, then there exists $c>0$ such that $K_{n}(y-x)\geq c$ for all
$x\in M$ and $y\in B(z,r)\cap S_{n}$, then

$\nu*\lambda_{n}(B(z,r)\cap S_{n})\geq c\nu(M)\mu(B(z,r)\cap S_{n}).$

Then $\gamma((\nu\ast\lambda_{n})_{|S_{n}})=1$ where
$(\nu\ast\lambda_{n})_{|S_{n}}$ is the restriction of $\nu\ast\lambda_{n}$ to
$S_{n}$. Finally, from
$(\nu\ast\lambda_{n})_{|S_{n}}\to\nu_{|S_{\varepsilon}}$ in total variation,
it follows that, for $n$ large enough,
$\|\nu-\nu\ast\lambda_{n_{|S_{n}}}\|_{TV}<2\varepsilon$. It remains to be
proved that if $\nu$ is such that $\gamma(\nu)=1$, then for all
$\varepsilon>0$ there exists $\nu_{\varepsilon}$ such that
$\|\nu_{\varepsilon}-\nu\|_{TV}<\varepsilon$ and
$\gamma(\nu_{\varepsilon})=0$. We proceed as in the proof that (11) cannot be
estimated. We first consider $B(x,r)$ such that $\nu(B(x,r))<\varepsilon$ (it
can be $\nu(B(x,r))=0$), then we consider a non-empty segment on the ball and
define the uniform on that segment. The measure $\lambda$ defined as in
Theorem 6.2 fulfills that $\gamma(\lambda)=0$ and
$\|\lambda-\nu\|_{TV}<\varepsilon$. ∎

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