# 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 language English 35 Published - 26 Mar 2019 Yes

### 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

@techreport{81602e6bcd474d18a58e6b4f17a377e8,
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.",
keywords = "U-stqtistics, regression-type models, conditional distribution, Penalization method",
author = "Alexis Derumigny",
year = "2019",
month = "3",
day = "26",
language = "English",
type = "WorkingPaper",

}

2019.

Research output: Working paperProfessional

TY - UNPB

T1 - Estimation of a regular conditional functional by conditional U-statistics regression

AU - Derumigny, Alexis

PY - 2019/3/26

Y1 - 2019/3/26

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.

KW - U-stqtistics

KW - regression-type models

KW - conditional distribution

KW - Penalization method

M3 - Working paper

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

ER -