On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

Salim Bouzebda

Makale detayları

On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

Yazarlar:

Salim Bouzebda

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistics

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 5, pp.1303-1348

Belge Tipi:

Makale

Yayın Tarihi:

10/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301692

pISSN:

2651-477X

Anahtar kelimeler:

Özet:

 U-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. U-statistics generalise the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute Conditional U -statistics, Ann. Probab., 1991 introduced a class of estimators called conditional U-statistics. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over varphi in mathscrF for a suitably restricted class mathscrF, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric Inverse Probability of Censoring Weighted estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional U-statistics processes. We discuss the wild bootstrap of the conditional U-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.

 U-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. U-statistics generalise the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute Conditional U -statistics, Ann. Probab., 1991 introduced a class of estimators called conditional U-statistics. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over varphi in mathscrF for a suitably restricted class mathscrF, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric Inverse Probability of Censoring Weighted estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional U-statistics processes. We discuss the wild bootstrap of the conditional U-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.

 U-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. U-statistics generalise the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute Conditional U -statistics, Ann. Probab., 1991 introduced a class of estimators called conditional U-statistics. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over varphi in mathscrF for a suitably restricted class mathscrF, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric Inverse Probability of Censoring Weighted estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional U-statistics processes. We discuss the wild bootstrap of the conditional U-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.

 U-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. U-statistics generalise the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute Conditional U -statistics, Ann. Probab., 1991 introduced a class of estimators called conditional U-statistics. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over varphi in mathscrF for a suitably restricted class mathscrF, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. Moreover, in the same context, we show the uniform bandwidth consistency for the nonparametric Inverse Probability of Censoring Weighted estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional U-statistics processes. We discuss the wild bootstrap of the conditional U-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.

Makale detayları

On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

Yazarlar:

Salim Bouzebda

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistics

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 5, pp.1303-1348

Belge Tipi:

Makale

Yayın Tarihi:

10/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301692

pISSN:

2651-477X

Conditional U-statistics

consistency

data-dependent bandwidth selection

empirical process

kernel estimation

Nadaraya-Watson

regression

copula function

uniform in bandwidth

weak convergence

wild bootstrap

Sayı

Makale detayları

Yazarlar: Salim Bouzebda

Yayıncı: Hacettepe University

Ana Başlıklar: Statistics

Dergi: Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 5, pp.1303-1348

Belge Tipi: Makale

Yayın Tarihi: 10/2023

Belge Dili: İngilizce

Document ID (Org No.): 2001301692

pISSN: 2651-477X

Sayı

Yazarlar:

Salim Bouzebda

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistics

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.52 Issue 5, pp.1303-1348

Belge Tipi:

Makale

Yayın Tarihi:

10/2023

Belge Dili:

İngilizce

Document ID (Org No.):

2001301692

pISSN:

2651-477X