Functional, randomized and smoothed multivariate quantile regions

Faugeras, Olivier; Rüschendorf, Ludger

doi:10.1016/j.jmva.2021.104802

Cited by 1 publication

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“…The introduction by of center-outward ranks and quantiles rapidly attracted the attention of the nonparametric community. It has triggered, among others, Faugeras and Rüschendorf (Faugeras and Rüschendorf (2017); Faugeras and Rüschendorf (2019)) and de Valk and Segers de Valk and Segers (2018). Applications to the long-standing open problem of constructing distribution-free tests for the hypothesis of independence between vectors with unspecified densities have been proposed by ), Shi, Drton, and Han Shi et al (2019), Shi, Hallin, Drton, and Han Shi et al (2020, and Ghosal and Sen .…”

Section: Introduction: Center-outward Distribution and Quantile Funct...mentioning

confidence: 99%

Statistical analysis of the optimal transport problem

González Sanz

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Optimal transportation is a resource allocation problem present in fields such as economics, finance, physics or artificial intelligence. From a probabilistic point of view, the optimal transport cost endows the space of probability measures with a metric topology. In particular, this topology is equivalent to the weak topology of probability measures together with the convergence of moments. This makes the transport cost an appropriate tool for measuring discrepancies between distributions. On the other hand, the solution of the transport problem is known as optimal plan. That is, an unambiguous way to relate two distributions following an optimality criterion. This optimal plan, when deterministic, is called a transport map.However, in many cases the probability distribution is a theoretical, unattainable entity. It is only visible to the practitioner through its empirical version, i.e. a finite data set of size n. This work examines the asymptotic behaviour of the transport cost in its empirical version.In other words, we study the limits of the empirical cost and plans when the data grows to infinity. It is well-known that the empirical transport cost converges to the population one. Moreover, for continuous measures it does so at a rate that decreases with dimension. In this thesis we prove the consistency of the transport map using topology of set-valued maps. This leads, indirectly, to being able to state that the rate at which the fluctuations-difference between the expected empirical cost and the empirical cost itself-approximate zero is the parametric n − 1 2 , irrespective of the dimension. Moreover, these fluctuations multiplied by n 1 2 RésuméLe transport optimal est un problème d'allocation de ressources que l'on retrouve dans des domaines tels que l'économie, la finance, la physique et l'intelligence artificielle. D'un point de vue probabiliste, le coût de transport optimal dote l'espace des mesures de probabilité d'une topologie métrique. Cela fait du coût de transport un outil approprié pour mesurer les écarts entre les distributions. D'autre part, la solution du problème de transport est connue comme le plan optimal. C'est-à-dire une manière non ambiguë de mettre en relation deux distributions suivant un critère d'optimalité. Ce plan optimal, lorsqu'il est déterministe, s'appelle une application de transport. Cependant, la distribution de probabilité est souvent une entité théorique, irréalisable. Elle n'est visible pour le praticien qu'à travers sa version empirique, c'est-à-dire un ensemble de données fini de taille n. Ce document examine le comportement asymptotique du coût de transport dans sa version empirique. En d'autres termes, nous étudions les limites du coût empirique et de le plan lorsque les données croissent à l'infini. Les travaux précédents ont montré que le coût de transport empirique converge vers le coût théorique. De plus, pour les mesures continues, elle le fait à un taux qui diminue avec la dimension. Dans cette thèse, nous démontrons la cohérence de l'application de transp...

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Section: Introduction: Center-outward Distribution and Quantile Funct...mentioning

confidence: 99%

Statistical analysis of the optimal transport problem

González Sanz

View full text Add to dashboard Cite

show abstract

Functional, randomized and smoothed multivariate quantile regions

Cited by 1 publication

References 14 publications

Statistical analysis of the optimal transport problem

Statistical analysis of the optimal transport problem

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