A prelude to statistics in Wasserstein metric spaces

Chon Van Le; Uyen Hoang Pham Article A prelude to statistics in Wasserstein metric spaces Asian Journal of Economics and Banking (AJEB) Provided in Cooperation with: Ho Chi Minh University of Banking (HUB), Ho Chi Minh City Suggested Citation: Chon Van Le; Uyen Hoang Pham (2024) : A prelude to statistics in Wasserstein metric spaces, Asian Journal of Economics and Banking (AJEB), ISSN 2633-7991, Emerald, Leeds, Vol. 8, Iss. 1, pp. 54-66, https://doi.org/10.1108/AJEB-10-2023-0099 This Version is available at: https://hdl.handle.net/10419/334113 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ A prelude to statistics in Wasserstein metric spaces Chon Van Le International University, Vietnam National University –Ho Chi Minh City, Ho Chi Minh City, Vietnam, and Uyen Hoang Pham University of Economics and Law, Vietnam National University –Ho Chi Minh City, Ho Chi Minh City, Vietnam Abstract Purpose –This paper aims mainly at introducing applied statisticians and econometricians to the current research methodology with non-Euclidean data sets. Specifically, it provides the basis and rationale for statistics in Wasserstein space, where the metric on probability measures is taken as a Wasserstein metric arising from optimal transport theory. Design/methodology/approach –The authors spell out the basis and rationale for using Wasserstein metrics on the data space of (random) probability measures. Findings –In elaborating the new statistical analysis of non-Euclidean data sets, the paper illustrates the generalization of traditional aspects of statistical inference following Frechet’s program. Originality/value –Besides the elaboration of research methodology for a new data analysis, the paper discusses the applications of Wasserstein metrics to the robustness of financial risk measures. Keywords Frechet mean sets, Histogram data sets, Optimal transport, Random probability measures, Robustness of financial risk measures, Wasserstein metrics, Wasserstein sampling spaces, WGAN Paper type Research paper 1. Introduction As we are witnessing the current extension of statistical analysis to more general data sets in data science, it is about time to let applied statisticians and econometricians be aware of this useful and important phenomenon. The cornerstone of statistical theory for applications is data. Traditionally, data are elements of Euclidean spaces which are naturally equipped with Euclidean distances which are essential for analysis. Modern applications call for more general data sets, such as histograms or non-Euclidean data. To use statistics to make predictions and decisions with this new type of data, we need to extend traditional statistical theory. The first basic ingredient to generalize is metrics on new data sets. This short note aims simply at elaborating a bit on a popular new metric which applied econometricians can learn to apply to their empirical applications from current research literature. This popular new metric is called Wasserstein metric (distance) which is shown to be suitable for a variety of non-Euclidean data space, such as Wasserstein space which is a space of probability distributions equipped with a Wasserstein metric. Simple and elementary examples will serve as illustrating the usefulness and rationale of modern statistics with non-Euclidean data. The note elaborates theoretical aspects in simple AJEB 8,1 54 JEL Classification —C10 © Chon Van Le and Uyen Hoang Pham. Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons. org/licences/by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2615-9821.htm Received 8 October 2023 Revised 17 October 2023 19 October 2023 31 October 2023 Accepted 6 November 2023 Asian Journal of Economics and Banking Vol. 8 No. 1, 2024 pp. 54-66 Emerald Publishing Limited e-ISSN: 2633-7991 p-ISSN: 2615-9821 DOI 10.1108/AJEB-10-2023-0099 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 settings, as well as mentioning some concrete applications. Our purpose is simply introducing applied statisticians and econometricians to modern data analysis based upon statistical theory. The paper is organized as follows. In Section 2, we elaborate on Wasserstein metrics in a concrete data set consisting of (random) histograms which are probability measures, together with the notion of Wasserstein metrics. In Section 3, we touch upon the starting point to generalize traditional statistics in Euclidean spaces to Wasserstein spaces. In Section 4,we mention an application of Wasserstein metrics to the robustness issue of financial risk management. Section 5 provides the conclusions. 2. Wasserstein metrics on histogram data sets We can take it as self-evidence that statistics is based on data. While we do have a general theory of statistics to guide us each time we need statistics, there is something hidden in the practices of statistics that we start looking at nowadays. Traditionally, most of our data are Euclidean elements and in practicing statistics on Rk, we take for granted their Euclidean distances k.k k , without bothering spelling out that our data set is a metric space (which is, in fact, essential for all statistical investigations, such as comparing data points, summarizing observed data sample). Before our times, i.e. before we actually run into modern applications where our data could be non-Euclidean, Maurice Frechet has forseen the future (i.e. nowadays) for us. Indeed, recognizing that our traditional data space is the metric space ðRd; :kdÞ,Frechet (1906) first axiomatized the notion of a metric on arbitrary spaces, to have rigorous metric spaces, not only for mathematical functional calculus, but specifically for probability and statistics. A well-known situation for all statisticians where “data points”are non-Euclidean is this. Let X 1 ,X 2 ,...,X n be an observed (IID) random sample drawn from a real-valued random variable (population) Xwhose distribution function Fis unknown. To improve the classical practices (e.g. estimating some population parameters of interest), and to take into account the advantages of computer science, the method of bootstrap was invented to improve the accuracy of estimators and their confidence intervals. The method consists of creating new “data points”via simulations. Specifically, given the observed sample X 1 ,X 2 ,...,X n , we obtain the known empirical distribution function (but, ex ante, it is a random distribution function): FnðxÞ¼1 nX n j¼1 1ð−∞;xðXjÞ whose corresponding probability measure (law) is dFnð:Þ¼1 nPn j¼1vXjð:Þ(by LebesgueStieltjes Theorem) where vXjð:Þ¼1ð:ÞðXjÞis the (random) Dirac probability measure at X j on BðRÞ. Having the known probability measure dF n , we can create simulated data from it via F−1 nðUÞ, where F−1 nð:Þ:½0;1→R;F−1 nðuÞ¼inffx∈R:FnðxÞ≥ug is the (univariate) quantile function of F n and Uis the random variable uniformly distributed on [0,1]. Roughly speaking, a simulated sample (a new “data point”) is obtained as a result of drawing with replacement npoints from the set {X 1 ,X 2 ,...,X n }, say, mtimes, resulting in m sets B k 5{b 1,k ,b 2,k ,...,b n,k }, k51, 2, ...,m. Wasserstein metric spaces 55 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 Because of the drawings with replacement, the elements b j,k in each B k could be equal, i.e. appearing more than once in it, so that each new “data point”B k is not really a subset of n elements of Ras in set theory. Instead, each B k is a multiset, i.e. a collection of points distinct or not (multiplicities of occurences are allowed). As a remark, such a collection of npoints b j,k ,j51, 2, ...,n, can be viewed as a fuzzy subset of R, consisting of distinct points whose degrees of membership are equal to the ratios of their multiplicity of occurence and the size n. But, in the setting of statistics, it is more representative if we view the new “data points”B k as a histogram (a random probability measure on BðRÞ), so that our new data set is a space of (random) probability measures denoted as PðRÞwhere each “data point”is not an element of the Euclidean space R, but is a probability measure on the metric space ðR;j:jÞ. Data sets which are (random) probability measures on a metric space ðX; ρ Þabound in applications. As such, we need a suitable metric between probability measures. Remark. But we know well that a large part of probability theory was about precisely the metrization of weak convergence of probability measures on metric spaces, i.e. producing metrics on the space of probability measures, see, e.g. Billingsley (1995),Parthasarathy (1967). Can we just pick some known metric among, say, Levy, Prokhorov, Total Variation metrics to use? Well, it depends on what we want our chosen metric to “behave!”So far, metrics on probability measures are invented to study asymptotic sampling distributions, such as in the Central Limit Theorem. They were not invented to handle data analysis, in which we need, for example, to use a suitable metric to compare probability measures (as data points in our new data set of an application). For example, if we observe three data “points”as three probability densities f,g,hwhich are uniformly distributed on [3, 2], [2, 1] (Bernton et al., 2019; Bhat and Prashanth, 2019), respectively, (and denoting Fas the distribution function with density fand dF its associated probability measure) then TVðdF;dGÞ¼1 2Z∞ −∞ jfðxÞgðxÞjdx ¼1¼TVðdF;dHÞ i.e. the total variation metric cannot capture the locations of these histogram data. This is similar to the recognition that Hausdorff distance on subsets of a space cannot be used when data are curved in the space, although curves are subsets. The reason is clear: Hausdorff distance does not capture the structure of curves which is needed in data analysis when curves are data “points”. So, what are other metrics (on space of probability measures) which can be used for data analysis/statistics with data sets as histograms? We need to compare histograms (as data points) in applications when each histogram represents the observed information about an “object”, or the return of a stock in financial econometrics. Then it is obvious that we must take into account of their locations! On the other hand, if data points are elements of an Euclidean space, e.g. x;y∈ðR;j:jÞ, the suitable metric Wwe wish to have should be a natural extension of the Euclidean metric j.jon R, in the sense that W(v x ,v y )5jxyj, i.e. when we identify a number x∈Rwith the Dirac probability measure v x . We are going to “mention”a suitable and popular metric W(., .) on histogram data. It seems important for applied statisticians and econometricians to have a good understanding of that metric to feel comfortable to use it in real-world applications, rather than just take it for granted!. The following elaboration is for this purpose. As far as history is concerned, it is fair to start with Maurice Frechet, the pioneer of modern statistics. In 1937, Levy (1937) defined several metrics on probability measures on BðRÞ. One is AJEB 8,1 56 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 LðF;GÞ¼inff ε >0:Gðx ε Þ ε ≤FðxÞ≤Gðxþ ε Þþ ε ; ∇ x∈Rg which metrized the convergence in distribution (or weak convergence of probability measures, i.e. Fn→ wFif F n (x)→F(x), as n→∞, for any x∈C(F), the continuity set of F(.)) i.e. Fn→ wF5LðFn;FÞ→0. Pursuing Levy’s work, in 1957, Frechet (1957) observed that Levy’s distance L(F,G) of the distribution functions of two random variables Xand Yinvolved Fand Galone. He suggested that a “global”distance W H (X,Y) should involve the joint distribution function H(x,y) of the random vector (X,Y), say, W H (F,G) where Hð:; :Þ:R2→½0;1is the joint distribution with marginals F,G, i.e. H(x,∞)5F(x), H(∞,y)5G(y). Another definition of Levy’s distance on distribution functions on Ris of the form WðF;GÞ¼inffWHðF;GÞ:H∈CðF;GÞg where C(F,G) is the set of joint distributions with marginals F,G(later in 1959, Abe Sklar specified it as copulas). But for W(F,G) to be a bona fide “metric”(in particular, W(F,G)5F5G), the above infimum must be attained at some special H*. Let’s see whether it is the case or not for the example given in Frechet (1957) WHðF;GÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EHðXYÞ2 q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ZR2 ðxyÞ2dHðx;yÞ s Remark. In 1969, Vassershtein (Wasserstein) (1969) proposed exactly W1ð μ ; ν Þ¼inf ZR2 xy dλðx;yÞ:λ∈Πð μ ; ν Þ  where Π( μ , ν ) is the set of probability measures on BðR2Þwith projections (marginals) μ , ν . Upfront: inf ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EHðX−YÞ2 q:H∈CðF;GÞg is attained at H*(x,y)5F(x)∧G(y) because, for Uuniformly distributed on [0, 1], X¼ DF−1ðUÞ,Y¼ DG−1ðUÞ, the joint distribution function of (F 1 (U), G 1 (U)) is H* and WH*ðF;GÞ¼WH*F−1ðUÞ;G−1ðUÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z1 0F−1ðuÞG−1ðuÞ2 du s which is the minimum of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EHðX−YÞ2 q:H∈CðF;GÞ  . It suffices to show that W1ðF;GÞ¼inffRR2x−ydHðx;yÞ:H∈CðF;GÞg is attained at H*(x,y)5F(x)∧G(y). The same result holds for W p ,p≥1, where WpðF;GÞ¼ inf ZR2 xyjpdHðx;yÞ:H∈CðF;GÞ  1 p Here are the details, see Vallender (1973), that the infimum of RR2jx−yjdHðx;yÞover H∈C(F, G) is indeed attained (at H(x,y)5F(x)∧G(y)). Let X;Y:ðΩ;A;PÞ→ðR;BðRÞÞ be random variables with distributions F,G, respectively. Wasserstein metric spaces 57 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 Since jXYj¼ðXYÞ1ðX≥YÞþðYXÞ1ðX<YÞ we let α 5max(XY, 0) and β5max(YX), 0), so that EjXYj¼E α þEβ Since α ≥0, we have Eð α jY¼yÞ¼Z∞ 0 Pð α >zjY¼yÞdz Now, E( α )5EE( α jY), so that E α ¼Z∞ −∞ dGðyÞZ∞ 0 PðXY≥zjY¼yÞ¼ Z∞ −∞ dGðyÞZ∞ 0 PðX≥yþzjY¼yÞdz ¼ Z∞ −∞ dGðyÞZ∞ y PðX≥xjY¼yÞdx ¼ ZZðx;yÞ;x>y PðX≥y;Y<yÞdx ¼Z∞ −∞ PðX≥y;Y<yÞdy Similarly, Eβ¼Z∞ −∞ PðY≥y;X<yÞdy Thus, EjXYj¼Z∞ −∞ PðX≥y;Y<yÞdy þZ∞ −∞ PðY≥y;X<yÞdy ¼ Z∞ −∞ ½PðX<y;Y≥yÞþPðY<y;X≥yÞdy Now, look at the event (X<y,Y≥y). Let A5(X<y) and B5(Y<y), then (X<y,Y≥y)5A∩B c . But A5(B c ∩A)∪(A∩B), so that PðX<y;Y≥yÞ¼PðAÞPðA\BÞ¼PðX<yÞPðX<y;Y<yÞ Thus, EjXYj¼Z∞ −∞ ½PðX<yÞþPðY<yÞ2PðX<y;Y<yÞdy Note that P(X<y,Y<y) is the value of the joint distribution H(y,y) of the vector (X,Y), and it is well known that H(x,y)≤F(x)∧G(y) which is a joint distribution with marginals F,G(from Frechet’s work (1956) on correlation analysis with given marginals or from copula theory), so that AJEB 8,1 58 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 EjXYj≥Z∞ −∞ ½FðyÞþGðyÞ2min FððyÞ;GðyÞdy ¼Z∞ −∞ jFðyÞGðyÞjdy by noting that jxyj5xþy2(x∧y). Therefore, W1ðdF;dGÞ¼inffWHðF;GÞ:H∈CðF;GÞg ≥Z∞ −∞ jFðyÞGðyÞjdy But Z∞ −∞ jFðyÞGðyÞjdy ¼Z1 0 jF−1ðuÞG−1ðuÞjdu (by an “analytic”proof below) so that the infimum of RR2jx−yjdHðx;yÞover H∈C(F,G)is R1 0jF−1ðuÞ−G−1ðuÞjdu which turns out to be a minimum since Z1 0 jF−1ðuÞG−1ðuÞjdu ¼EjF−1ðUÞG−1ðUÞj ¼ EH*jXYj where H*(x,y)5F(x)∧G(y) is the joint distribution function of (F 1 (U), G 1 (U)). Q.E.D. Remarks. (1) Let X¼ DF½−1ðUÞand Y¼ DG½−1ðUÞ, we have dH*¼du◦ðF−1;G−1Þ−1, so that H*ðx;yÞ¼dH*ðð−∞;x3ð−∞;yÞ ¼ dunu:F−1ðuÞ≤x;G−1ðuÞ≤yo¼ dufu:u≤FðxÞ;u≤GðyÞg ¼ dufu:u≤FðxÞ∧GðyÞg ¼ FðxÞ∧GðyÞ (2) Proof of Z∞ −∞ jFðyÞGðyÞjdy ¼Z1 0 jF−1ðuÞG−1ðuÞjdu is as follows. The following is justified by Fubini’s theorem, namely if R A3B jf(x,y)jd(x, y)<∞, then ZA3B jfðx;yÞjdðx;yÞ¼ZAZB fðx;yÞdy  dx ¼ZBZA fðx;yÞdx  dy Now, for u∈(0, 1), we have jF½−1ðuÞG½−1ðuÞj ¼ hF½−1ðuÞG½−1ðuÞi1fu:F½−1ðuÞ>G½−1ðuÞgðuÞþ hG½−1ðuÞF½−1ðuÞi1fu:F½−1ðuÞ≤G½−1ðuÞgðuÞ So let A¼nu∈ð0;1Þ:F½−1ðuÞ>G½−1ðuÞo Wasserstein metric spaces 59 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 Ac¼nu∈ð0;1Þ:F½−1ðuÞ≤G½−1ðuÞo We have Z1 0 jF½−1ðuÞG½−1ðuÞjdu ¼ ZA jF½−1ðuÞG½−1jðuÞjdu þZAc jF½−1ðuÞG½−1ðuÞjdu where we can write ZA jF½−1ðuÞG½−1jðuÞjdu ¼ZAZF½−1ðuÞ G½−1ðuÞ dx "# du Now, observe that, by definition of the quantile functions, we have G [1] (u)≤x5u≤G(x) (and of course, x<F [1] (u)5u>F(x)), so that ZAZF½−1ðuÞ G½−1ðuÞ dx "# du ¼ZRZGðxÞ FðxÞ 1AðuÞ1fFðxÞ≤GðxÞgðxÞdu "# dx Similarly, ZAc jF½−1ðuÞG½−1ðuÞjdu ¼ZRZFðxÞ GðxÞ 1AcðuÞ1fFðxÞ>GðxÞgðxÞdu "# dx Hence, ZRZGðxÞ FðxÞ 1AðuÞ1fFðxÞ≤GðxÞgðxÞdu "# dx þZRZFðxÞ GðxÞ 1AcðuÞ1fFðxÞ>GðxÞgðxÞdu "# dx ¼ ZR jFðxÞGðxÞjdx Q.E.D. Now, the distance W 1 (F,G)orW1ðdF;dGÞ¼R1 0jF−1ðuÞ−G−1ðuÞjdu does take into account the locations of the histogram data “points”. Indeed, for the histograms f,g,hin the previous example (with associated distributions F,G,H, respectively), we have W 1 (F,G)51 and W 1 (F,H)55, showing that the histogram fis closer to gthan h. On the other hand, W 1 is a natural extension from Euclidean data points to histogram data points. Indeed, for x;y∈R, we identify them as vxðAÞ¼dFxðAÞ¼1AðxÞ;vyðBÞ¼¼dGyðBÞ¼1BðyÞ so that FxðtÞ¼vxðð−∞;tÞ ¼ 1½x;∞ÞðtÞ Since we consider real-valued random variable, i.e. with values in R¼ð−∞;∞Þ, their quantile functions, e.g. F−1 xð:Þ:ð0;1Þ→R: AJEB 8,1 60 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025 F−1 xðuÞ¼infft∈R:FxðtÞ≥ug¼x1ð0;1ÞðuÞ and hence W1ðvx;vyÞ¼Z1 0 jF−1 xðuÞF−1 yðuÞjdu ¼Z1 0 jxyj1ð0;1ÞðuÞdu ¼jxyj Remarks. (1) In 1969, Wasserstein Vassershtein (1969) considered W 2 to investigate the uniqueness of the stationary distribution of a Markov process. And in 1970, Dobrushin (1970) used Wasserstein metric to investigate stochastic processes by conditional distributions. (2) In 1972, Mallows (1972) considered the same W 2 metric, without referring to its existence years ago! (3) Shorack and Wellner (1986) used Wasserstein metrics to investigate the convergence of empirical processes in their book in 1986. (4) For general Wasserstein metrics in Optimal Transport Theory, see Villani (2003) 3. Typical positions in Frechet’s program From a historical perspective, the pioneering work of Frechet (1948) can be viewed as the first attempt to generalize probability background for statistics, such as general random elements in arbitrary metric spaces, their typical positions (e.g. mean), general parameters, general statistics and their convergences (for asymptotics, e.g. consistency of estimators). Nowadays, we are witnessing efforts of theoretical statisticians to specify Frechet’s vision while applied statisticians in various fields, such as economics and machine learning (ML), started by implementing it in real-world applications, see, e.g. Bernton et al. (2019),Bhat and Prashanth (2019),Bigot (2020),Chartier (2013) and Kiesel et al. (2016). We will elaborate on these current efforts in the context of Wasserstein metric spaces as data sets. For an invitation to the theoretical aspects of statistics in Wasserstein space, see Panaretos and Zemel (2020). Remark. As Breiman (2001) spelled out the useful marriage between statistics and ML, see, e.g. Morizet (2020),Shalev-Shwartz and Ben-David (2014), Wasserstein metrics are used also in ML, e.g. in WGAN. As a starting point, let’s discuss the notion of “typical positions”of a random element X with values in an arbitrary metric space ðX; ρ Þ. According to Frechet (1948), generalized typical positions such as median and mean could be defined via appropriate characterizations of classical notions on Euclidean spaces. For simplicity, consider ðR;j:jÞ. Let Xbe a real-valued random variable with distribution function F(and law dF). In classical probability theory, the median m(X)ofXis a value on Rwhich is “equiprobable” (always existed). The mean of Xis the quantity EX ¼RRxdFðxÞwhich exists when this integral is finite. To generalize these typical positions to arbitrary metric spaces, we need to “characterize”them. First, a characterization of m(X) is obtained when statisticians use LAD (Least Absolute Deviation) EjXajas error, say, in quantile regression. Wasserstein metric spaces 61 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/1/54/9525913/ajeb-10-2023-0099.pdf by ZBW German National Library of Economics user on 16 December 2025