1 Introduction: center-outward distribution and quantile functions
Univariate distribution and quantiles functions, together with their empirical counterparts and the closely related concepts of ranks and order statistics, count among the most fundamental and useful tools in mathematical statistics. Ranks indeed are not just distribution-free: in models driven by noise with unspecified density, they generate the sub--field of all distribution-free events (see ), which is also the largest sub--field independent, irrespective of the underlying distribution, of the minimal sufficient -field generated by the order statistic; suitable rank-based procedures achieve optimality in several senses in nonparametric testing as well semiparametric efficiency (see, e.g. , , , ). A major limitation of the classical concepts of ranks and quantiles, however, is that, due to the absence of a canonical ordering of for , they do not readily extend to the multivariate context.
The problem is not new, and numerous attempts have been made to fill that gap by defining multivariate versions of distribution and quantiles functions, with the ultimate goal of constructing suitable mutivariate versions of classical rank- and quantile-based inference procedures. The traditional definition of a multivariate distribution function is somewhat helpless in that respect, and does not produce any satisfactory concept of quantiles—let alone a satisfactory concept of ranks (see ). The componentwise approach, closely related with copula transforms, has been studied intensively (see ), but does not even enjoy distribution-freeness. Nor do the so-called spatial ranks (, ) inspired by the L characterization of univariate quantiles. The whole theory of statistical depth (see ,  for authoritative surveys), in a sense, is motivated by the same objective of providing a (data-driven) ordering of and adequate concepts of multivariate ranks () and quantiles (); here again, the resulting notions fail to be distribution-free. As for the Mahalanobis ranks and signs considered, e.g. in ,  or , they do enjoy distribution-freeness and all the desired properties expected from ranks—under the restrictive assumption, however, of elliptical symmetry.
This shortcoming of all available solutions has motivated the introduction, in  and , of the measure transportation-based concepts of Monge-Kantotovich depth, center-outward distribution and quantile functions, ranks, and signs. These center-outward concepts, unlike all previous ones, are shown (see , ) to enjoy all the properties that make their univariate counterpart a fundamental and successful tool for statistical inference; we refer to  for more references and further discussion.
Letand denote by
the uniform distribution on: then is a solution to Monge’s quadratic transportation problem, that is,
(see, e.g., ) where denotes the push forward of by —namely, the distribution of under ( a measurable map from to ). With generalization to higher dimension in mind, however,  and  rather consider , the so-called center-outward distribution function of , satisfying the transportation problem
where is the uniform distribution over , the one-dimensional unit ball . Clearly, and carry the same information about .
The latter definition, indeed, readily extends to arbitrary dimensions. Let denote a Borel probability measure on with finite second-order moments and Lebesgue density . Measure transportation theory (see, e.g., Theorem 2.12 in ) tells us that there exists a -a.s. unique map such that
where stands for the Euclidean norm of and denotes the uniform distribution over the open -dimensional unit ball . The center-outward distribution function is defined as a solution of this optimal transportation problem.
By uniform over we refer to spherical uniformity, that is, here corresponds to the uniform choice of a direction on the unit sphere in combined with an independent uniform choice in of a distance to the origin. A simple change of variable shows that has density
where denotes the area (the -dimensional Hausdorff measure, see, e.g., ) of the sphere . Note the singularity at the origin since is infinite at ; while we safely can neglect itself, which has measure zero, by putting , nevertheless remains unbounded in the vicinity of .
This definition of the center-outward distribution function as the solution of a quadratic transportation problem suffers from two major limitations. First, finite second-order moments are needed to make sense of the underlying optimization problem (1.1). Second, the distribution function based on (1.1) is only defined -a.s.; this means, for instance, that is not well defined outside the support of .
The first of these two limitations has been relaxed in  thanks to a celebrated theorem by McCann . Under the assumption that has finite second-order moments, Brenier in 1991 had shown that optimal transportation maps (hence, all versions of the -a.s. unique solution of Monge’s problem (1.1)) coincide -a.s. with the Lebesgue-a.e. gradient of a convex function222The notation here is used for the Lebesgue-a.e. gradient of , that is, is defined as the gradient at of whenever is differentiable at —which, for a convex , holds Lebesgue-a.e. Note that, contrary to , which is a.e. unique, is not—unless we impose, without loss of generality (see, e.g., Lemma 2.1 in ), that . , which has the interpretation of a potential. More precisely, a.s. is of the form where (i) is lower semicontinous (lsc in the sequel), (ii) is convex, and (iii) is such that . McCann  further showed that these last three conditions uniquely determine , even in the absence of second moment assumptions, while under finite second-order moments, is a solution of Monge’s problem (1.1). Thus, putting
the center-outward distribution function is no longer characterized as the almost surely unique solution of an optimization problem (1.1) requiring finite moments of order two but as the unique a.e. gradient of a convex function pushing forward to . We nevertheless conform to the common usage of improperly calling the optimal transport pushing forward to .
While taking care of the moment assumption—existence of second-order moments indeed is an embarrassing assumption when distribution and quantile functions are to be defined—the second limitation still remains. The non-unicity of , however, disappears if is such that is everywhere differentiable. That this is indeed the case was shown by Figalli in 2018  for in the so-called class of distributions with nonvanishing densities333Precisely, the distributions with densities and support satisfying Assumption A below.. For any in that class of distributions, Figalli actually establishes that is a gradient for all and, when restricted to
a homeomorphism between and the punctured ball . The latter property is quite essential if sensible—namely, closed, continuous, connected, and nested—quantile regions and contours, based on an inverse444See Section 2.1 for a precise definition. of , are to be defined: see  and .
The goal of this paper is to provide simple sufficient conditions for Figalli’s results to hold beyond the assumption of nonvanishing densities; we more particularly consider distributions with (bounded or unbounded) convex supports. Beyond other theoretical considerations, these are the key properties required to prove a.s. convergence of the empirical center-outward distribution functions to their theoretical counterparts (see ). Hence, the results of the present paper also are extending the validity of the center-outward Glivenko-Cantelli theorem in that reference.
From a technical point of view, our main result is Theorem 2.5 below, which relies on the classical regularity theory for solutions of Monge-Ampère equations associated with the name of of Caffarelli (see [4, 5, 6]), as discussed in Section 2. The use of that theory to investigate the regularity of optimal transportation maps between two probabilities typically requires that both probabilities have densities that are bounded and bounded away from zero over their respective supports. Recently, under a local version of this condition, a very general regularity result of this kind has been given in . However, the spherical uniform reference measure considered here, in dimension , yields unbounded densities at the origin, so that the results in  do not apply.555Note that the choice of the spherical uniform reference is not a whimsical one. It preserves the independence between and (extending the independence, for , between and sign) and produces simple and easily interpretable quantile contours with prescribed probability content (we refer to  for details). To our knowledge, the only reference dealing with this kind of unbounded density is  which, however, requires to be supported on the whole space . Here we extend the result in  to cover the case of with (bounded or unbounded) convex supports.
The sequel of this paper is organized as follows. Our main regularity result is established in Section 2, along with a succint account of the main elements of Cafarelli’s theory and some auxiliary results. We conclude with Section 3, which presents some new results on center-outward distribution and quantile functions. These include an asymptotic invariance property extending a well-known feature of classical univariate distribution functions and the ability of quantile contours to capture the shape of the bounded support of a probability measure by convergencing (in Hausdorff distance) to the boundary of the support. Finally, we include a result on the geometry of quantile sets, showing that they turn out to exhibit a limiting form of “lighthouse convexity”.
2 Regularity of center-outward distribution and quantile functions
2.1 Center-outward quantile functions
The Introduction was focused on the distribution functions . Exchanging the roles of and , we could have emphasized transportation from the unit ball to the support of , leading to the definition of the center-outward quantile function with, mutatis mutandis, the same comments.
Let denote a Borel probability measure over with Lebesgue density . While the center-outward distribution function is defined as the optimal (in the McCann sense) transport pushing forward to , the center-outward quantile map or quantile function of is defined as the optimal transport pushing forward to . Namely,
where is, in agreement with McCann’s Theorem, the unique a.e. gradient of a convex function with domain containing 666We adhere to the usual convention of considering that a function defined on is convex if it can be extended to a convex function on with values in ; the domain of the convex function is then redefined as the set where it takes finite values. such that . Again, imposing, without loss of generality,777Indeed, two convex functions with a.e. equal gradients on an open convex set are equal up to an additive constant (see, e.g., Lemma 2.1 in ). that , the convex potential is uniquely defined and a.e. differentiable over . We extend to a lsc convex function on with the standard procedure of setting if and for (see, e.g. (A.18) in ). With this extension, is the Legendre transform of , that is,
We observe that the domain of is and that , being the sup of a 1-Lipschitz function, is also 1 - Lipschitz. In particular, for almost every , is differentiable with and, as a consequence (see, e.g., Corollary A.27 in ),
here, and throughout this paper, stands for the closure of a set , for the subdifferential888Recall that the subdifferential of at is the set of all such that for all . of the convex function at , and . Furthermore, Proposition 10 in  (see also Remark 16) shows that, since has a density, for almost every in the support of and for almost every . In that sense, and are the inverse of each other. In this way, we have defined for almost every and for almost every ; the definitions coincide with those in  or  for in the support of .
2.2 Some regularity results for Monge-Ampère equations
As announced in the Introduction, our approach to the regularity of the center-outward distribution and quantile functions is based on the classical regularity theory for Monge-Ampère equations. We refer to  for a comprehensive account of this theory, of which we present here a minimal account.
Given an open set and a (finite) convex function , denoting by he Lebesgue measure on , the Monge-Ampère measure associated with is defined by
for every Borel set . It can be checked that is indeed a locally finite Borel measure on . The crucial link between center-outward distribution functions and Monge-Ampère measures can be summarized as follows. Assume is a probability on with Lebesgue density and let be a convex function from to . Then, for every Borel set ,
where is the Legendre transform of . We recall that convexity of implies that it is differentiable at almost every point in (see, e.g., Theorem 25.4 in ) and, therefore,
This and the fact that if and only if yield the last equality above. Hence, if has a density , for every Borel set ,
(see Lemma 4.6 in ); if, moreover, ,
with as in (1.2). Thus, the Monge-Ampère measure is Lebesgue-absolutely continuous. Since the density of the absolutely continuous part of the Monge-Ampère measure is given by (see McCann  or Theorem 4.8 in ), we conclude that, for every Borel set ,
Let us focus now on the Monge-Ampère measure associated (see Section 2.1) with and (both defined over ). Since pushes forward to , we have that -a.e. in . By continuity (see Theorem 25.5 in ), for every point of differentiability of . Using again Corollary A.27 in , we conclude that is included in the convex hull of . Hence, if itself is convex, we obtain that
Analogous to (2.4), we have that
Now, denoting by the open ball with radius centered at the origin, let us assume that the Borel set , with , has Lebesgue measure zero. Since is compact, also is compact (see, e.g. Lemma A.22 in ). Hence, there exists such that
The following assumption, which requires the density of to be bounded and bounded away from 0 on compact subsets of the support, is absolutely essential (the same assumption is also made by Figalli in ).
Assumption A. For every , there exist constants such that
Since is convex (hence ), Assumption A entails
We summarize this discussion in the next proposition.
Let be a probability measure with density supported on the open set . Denote by the convex, lower semicontinuous function satisfying and and let be defined as in (2.2). Then,
is absolutely continuous with respect to and, for every Borel ,
if, moreover, is convex and satisfies Assumption A, then is absolutely continuous with respect to and, for every Borel set ,
Next, let us show that, for well-behaved probability measures (those with convex support and density satisfying Assumption A), the center-outward distribution function cannot map points in the interior of the support of to extremal points of the unit ball.
Let be a probability measure with density supported on the convex open set and such that Assumption A holds. Then , where .
Assume that there exists such that for some . Without loss of generality, we can assume . Since is open, there exists such that . For small , consider the sets
Now, if and , the monotonicity of implies that . Hence,
This shows that . But the density , inside , is bounded from below by and the density is bounded from above by inside for : then, in view of the transport equation (2.4), we have
This, however, cannot hold true since and as . The claim follows. ∎
We now proceed to provide sufficient conditions under which the center-outward quantile function is continuous at every point in the open unit ball (except, possibly, at the origin). It is well known that differentiability of a lower semicontinuous convex function (which entails continuity of its gradient) is equivalent to strict convexity of its convex conjugate (see Theorem 26.3 in ). As announced, the techniques we are using here are in the spirit of those developed by Caffarelli in ,  or Figalli in , 
, which in turn largely rely on the fact that, under some control for the Monge-Ampère measure, the intersection between the graph and supporting hyperplanes ofeither consists of a single point or has an extreme point (see Theorem 4.10 in ). A central result in Caffarelli’s regularity theory (see Corollary 4.21 in ) is that a strictly convex function on an open set for which there exist constants such that
for every Borel set is automatically of class for some that depends only on , and (condition (2.10) in the sequel will be summarized, with a slight abuse of notation, as ). The fact that for has an unbounded density adds some complication to the particular problem here, though. On the other hand, the density is bounded away from 0, which allows to control the growth of the Monge-Ampère measure, as we show next.
If satisfies the assumptions in Proposition 2.1(ii), denoting by a compact subset of , there exist constants and such that, for every Borel set ,
The compactness of entails that of ; in particular, for some . Hence, using Proposition 2.1(ii) and taking as in Assumption A, we obtain
For the upper bound in (2.11), note that the ball (where denotes the volume of the -dimensional unit ball) maximizes among all subsets of with Lebesgue measure . On the other hand, by the co-area formula (see, e.g., Proposition 1, p. 118 in ),
Note that the lower bound in Lemma 2.3 remains valid for a compact subset of provided that is bounded: indeed, that lower bound only requires the upper bound from Assumption A. A similar conclusion holds for the upper bound. Additionally, if the density is uniformly bounded, the lower bound holds for any subset of .
2.3 Main result
We are ready now for the main result of this note. Our proof follows the lines of , , and , but we cover cases in which the range of the extension to the whole space of the center-outward quantile function is not necessarily equal to . As in the last reference, we have to handle carefully the fact that is not necessarily bounded and use a “minimal” extension of the quantile function potential, namely,
Obviously, is still a lower semicontinuous convex function and coincides with for . Since for every differentiability point of in , we see (using, once more, Corollary A.27 in ) that, provided that is convex, . The “minimality” of the extension (2.13) refers to the fact that , as can be checked from a simple application of the Hahn-Banach separation theorem. Of course, the values of outside are not relevant for the study of its differentiability inside , but the use of will be useful in the next proof. We note also that the discussion leading to Proposition 2.1 can be reproduced with substituted for to conclude that is absolutely continuous with respect to the Lebesgue measure and that, for every Borel set ,
The main result of this note follows from the following crucial lemma.
Under the assumptions of Theorem 2.5, is strictly convex on .
To prove this, assume that the contrary holds true. Then, there exists and such that, putting , the convex set is not a singleton. By subtracting an affine function, we can assume and for all ; then, , which is closed since is lower semicontinuous. Also, by adding the convex function (note that on ), we can assume that . Being compact and convex, equals the closed convex hull of its extreme points; as a consequence, it must have at least two exposed points (otherwise it would be empty or a singleton). Let be one of them. If , we consider a small ball , say, around , such that . Then is a compact set, and hence for some . By Proposition 2.1(ii), we have constants such that the Monge-Ampère measure satisfies in . But the set has an exposed point in and this contradicts Theorem 4.10 in . Consequently, we must assume that . Observe that , hence . First consider the case where . Let and denote, respectively the open and the closed ball of radius centered at . Then, for small enough, ; consequently, there exists some such that . For small enough, we further can ensure that .
Without any loss of generality, let us assume that where
stands for the first vector in the canonical basis of(we can use a rotation otherwise):
For small enough, we have
For such , defining
in the Hausdorff distance999Recall that, for , as . Hence, for small enough, the sets are bounded open convex subsets of the ball . By Lemma 2.3, there exists some such that
for every and small enough.
Next, fix and such that and consider the normalizing map —namely, the affine transformation that normalizes101010A convex set is said to be normalized if . For each open bounded convex set there exists a unique invertible affine transformation normalizing (this is John’s celebrated Lemma of convex analysis, see Lemma A.13 in ). We refer to as the normalizing map and to as the normalized version of . ; denote by the normalized solution in of with the boundary condition on ( is the convex map that has Monge-Ampère measure in and vanishes at the boundary of ; its existence and uniqueness is guaranteed, for instance, by Proposition 4.2 in ). Since , we have that and, therefore, the map satisfies
This implies that
We consider the sets . Now , a normalized set (it contains the unit ball and is contained in the ball of radius , the dimension of the Euclidean space). This implies that there exists a constant , depending only on such that (see Theorem 4.23 in  or Lemma 3 in )
In view of Lemma 2.3, the subsequent remark, and the fact that , we have that is lower bounded over , that is, there exists such that for every . This and (2.17) thus imply that is bounded from below on . It follows that, for some ,
This implies that, for small enough, no ball of radius can contain . As a consequence, there exists such that . Using Corollary A.23 in , we conclude that
for some . On the other hand, using Lemma 2.11 again to upper bound , we obtain
and, by the Alexandrov maximum principle (e.g. Theorem 2.8. in ), this implies that
This means that the same arguments as in the proof of Theorem 4.10 in  yield
which is a contradiction.
Finally, consider the case where the exposed point of belongs to ; here again, it can be assumed that and that the exposed point of is . We also can assume, without loss of generality, that . Hence, . For small we consider the sets
Let and . Then and, thanks to the monotonicity of , we have that for every , which entails (take ) and (take ). This means that and , from which we deduce that . Since, by Lemma 2.2, we have , it follows that . Also, since both and belong to , which is a convex set with nonempty interior, we can argue as in pp. 8-9 of , to conclude that for small enough. From the transport equation, we have that