Marginal M-quantile regression for multivariate dependent data

Merlo, Luca; Petrella, Lea; Salvati, Nicola; Tzavidis, Nikos

doi:10.1016/j.csda.2022.107500

Cited by 7 publications

(4 citation statements)

References 92 publications

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“…A nonparametric quantile regression model based on a directional form of Tukey depth is developed in Hallin et al (2015) but suffers the same lack of control over the probability contents of the quantile regions involved as the depth-based quantile concept itself. So does also the directional concept of M-quantiles proposed by Merlo et al (2022).…”

Section: Quantile Regression Single-and Multiple-outputmentioning

confidence: 99%

“…None of the earlier attempts to define multiple-output regression quantiles-neither the depth-based definitions in Hallin et al (2015) or Paindaveine and Šiman (2011), the directional concepts of marginal M-quantiles (Breckling and Chambers, 1988) considered by Merlo et al (2022), nor the measure-transportation-based approach proposed by Carlier, Chernozhukov and Galichon (2016)-is characterizing quantile regions that satisfy requirements (1.3) and (1.6).…”

Section: Quantile Regression Single-and Multiple-outputmentioning

confidence: 99%

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Nonparametric Multiple-Output Center-Outward Quantile Regression

Barrio¹,

Sanz²,

Hallin³

2022

Preprint

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Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. ( 2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours which are shown to consistently reconstruct their population versions in the Pompeiu-Hausdorff topology. Our method is entirely non-parametric and performs well in simulations including heteroskedasticity and nonlinear trends; its power as a data-analytic tool is illustrated on some real datasets.

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Section: Quantile Regression Single-and Multiple-outputmentioning

confidence: 99%

Section: Quantile Regression Single-and Multiple-outputmentioning

confidence: 99%

Nonparametric Multiple-Output Center-Outward Quantile Regression

Barrio¹,

Sanz²,

Hallin³

2022

Preprint

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“…In the last part of the manuscript, we follow the approach to multiple output regression with M-quantiles of Daouia and Paindaveine [10], Hallin et al [15,16], Merlo et al [17] with our distorted M-quantiles in order to present the notions of distorted M-quantile regression regions and conditional regions. The single output (distorted) M-quantile regression resembles the classical quantile regression introduced by Koenker and Basset [18], see also Koenker [19], while its extension to multi-output models is based on univariate projections of the response variable.…”

Section: Introductionmentioning

confidence: 99%

Data Depth and Multiple Output Regression, the Distorted M-Quantiles Approach

Ochoa

Cascos

2022

Mathematics

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For a univariate distribution, its M-quantiles are obtained as solutions to asymmetric minimization problems dealing with the distance of a random variable to a fixed point. The asymmetry refers to the different weights awarded to the values of the random variable at either side of the fixed point. We focus on M-quantiles whose associated losses are given in terms of a power. In this setting, the classical quantiles are obtained for the first power, while the expectiles correspond to quadratic losses. The M-quantiles considered here are computed over distorted distributions, which allows to tune the weight awarded to the more central or peripheral parts of the distribution. These distorted M-quantiles are used in the multivariate setting to introduce novel families of central regions and their associated depth functions, which are further extended to the multiple output regression setting in the form of conditional and regression regions and conditional depths.

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“…; with a slight abuse of language, also call T ± (0) the regression median of Y with respect to X. None of the earlier attempts to define multiple-output regression quantiles-neither the depth-based definitions in Hallin et al (2015), the directional concepts of marginal M-quantiles (Breckling and Chambers, 1988) considered by Merlo et al (2022), nor the measure-transportation-based approach proposed by Carlier et al ( 2016)-is characterizing quantile regions that satisfy requirements (7.3) and (7.6). Carlier et al (2016) deserves special attention, though, as the first attempt to break with directional and depth-based approaches to multiple-output quantile regression by means of innovative measure transportation ideas.…”

Section: Quantile Regression Single-and Multiple-outputmentioning

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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Marginal M-quantile regression for multivariate dependent data

Cited by 7 publications

References 92 publications

Nonparametric Multiple-Output Center-Outward Quantile Regression

Nonparametric Multiple-Output Center-Outward Quantile Regression

Data Depth and Multiple Output Regression, the Distorted M-Quantiles Approach

Statistical analysis of the optimal transport problem

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