Estimation of a regular conditional functional by conditional U-statistics regression

Research output: Working paperProfessional

Abstract

U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable $X$ to sums over every $k$-tuple of distinct observations of $X$. They may be used to estimate a regular functional $\theta(P_{X})$ of the law of $X$. When a vector of covariates $Z$ is available, a conditional U-statistic may describe the effect of $z$ on the conditional law of $X$ given $Z=z$, by estimating a regular conditional functional $\theta(P_{X|Z=\cdot})$. We prove concentration inequalities for conditional U-statistics. Assuming a parametric model of the conditional functional of interest, we propose a regression-type estimator based on conditional U-statistics. Its theoretical properties are derived, first in a non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.
Original languageEnglish
Number of pages35
Publication statusPublished - 26 Mar 2019
Externally publishedYes

Fingerprint

U-statistics
Regression
Mean of a random variable
Concentration Inequalities
Estimator
Parametric Model
Covariates
Distinct
Estimate

Keywords

  • U-stqtistics
  • regression-type models
  • conditional distribution
  • Penalization method

Cite this

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title = "Estimation of a regular conditional functional by conditional U-statistics regression",
abstract = "U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable $X$ to sums over every $k$-tuple of distinct observations of $X$. They may be used to estimate a regular functional $\theta(P_{X})$ of the law of $X$. When a vector of covariates $Z$ is available, a conditional U-statistic may describe the effect of $z$ on the conditional law of $X$ given $Z=z$, by estimating a regular conditional functional $\theta(P_{X|Z=\cdot})$. We prove concentration inequalities for conditional U-statistics. Assuming a parametric model of the conditional functional of interest, we propose a regression-type estimator based on conditional U-statistics. Its theoretical properties are derived, first in a non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.",
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N2 - U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable $X$ to sums over every $k$-tuple of distinct observations of $X$. They may be used to estimate a regular functional $\theta(P_{X})$ of the law of $X$. When a vector of covariates $Z$ is available, a conditional U-statistic may describe the effect of $z$ on the conditional law of $X$ given $Z=z$, by estimating a regular conditional functional $\theta(P_{X|Z=\cdot})$. We prove concentration inequalities for conditional U-statistics. Assuming a parametric model of the conditional functional of interest, we propose a regression-type estimator based on conditional U-statistics. Its theoretical properties are derived, first in a non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.

AB - U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable $X$ to sums over every $k$-tuple of distinct observations of $X$. They may be used to estimate a regular functional $\theta(P_{X})$ of the law of $X$. When a vector of covariates $Z$ is available, a conditional U-statistic may describe the effect of $z$ on the conditional law of $X$ given $Z=z$, by estimating a regular conditional functional $\theta(P_{X|Z=\cdot})$. We prove concentration inequalities for conditional U-statistics. Assuming a parametric model of the conditional functional of interest, we propose a regression-type estimator based on conditional U-statistics. Its theoretical properties are derived, first in a non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.

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KW - regression-type models

KW - conditional distribution

KW - Penalization method

M3 - Working paper

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