### Abstract

Original language | English |
---|---|

Number of pages | 35 |

Publication status | Published - 26 Mar 2019 |

Externally published | Yes |

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### Keywords

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

### Cite this

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**Estimation of a regular conditional functional by conditional U-statistics regression.** / Derumigny, Alexis.

Research output: Working paper

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 -