Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy

In this study, we look at the wavelet basis for nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that it is ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination.

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“…We first decompose H T ( f ) − EH T ( f ), as in [92][93][94][95], into the sum of two components, by writing…”

Section: Shannon's Entropymentioning

confidence: 99%

Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

Didi

Bouzebda

2022

Mathematics

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“…Remark 4.3 The main problem in using entropy estimates such as in (1.5) is to choose properly the smoothing parameter h n . With a lot more effort, we could derive analog results here for H ε;n (X) using the methods in Bouzebda and Elhattab (2009, 2010, 2011, as well as the modern empirical process tools developed in Einmahl and Mason (2005) in their work on uniform in bandwidth consistency of kernel-type estimators.…”

Section: A Comparison Study By Simulationsmentioning

confidence: 99%

New Entropy Estimator with an Application to Test of Normality

Bouzebda

Elhattab

Keziou

et al. 2013

Communications in Statistics - Theory and Methods

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“…Nolan and Pollard [110] were the first to introduce the notion of uniform in bandwidth consistency for kernel density estimators and they applied empirical process methods in their study. In the series of papers, [11,18,19,20,21,23,28,30,45,52,53,56,61,62,105], the authors established uniform consistency results for such estimators, where h n varies within suitably chosen intervals indexed by n. U -statistics, first considered by [81] in connection with unbiased statistics, and formally introduced by [84]. The theory of Ustatistics and U -processes has received considerable attention in the last decades due to its great number of applications and usefulness for solving complex statistical problems.…”

Section: Introductionmentioning

confidence: 99%

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

BOUZEBDA

2023

Hacettepe Journal of Mathematics and Statistics

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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 φ ∈ Ƒ for a suitably restricted class Ƒ, 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.

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Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy

Cited by 13 publications

References 13 publications

Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

New Entropy Estimator with an Application to Test of Normality

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

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