Kernel Estimation of Cumulative Residual Tsallis Entropy and Its Dynamic Version under ρ-Mixing Dependent Data.

Muhammed Rasheed Irshad,Radhakumari Maya,Francesco Buono,Maria Longobardi

doi:10.3390/e24010009

Muhammed Rasheed Irshad, Radhakumari Maya + Show 2 more

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https://doi.org/10.3390/e24010009

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Abstract

Tsallis introduced a non-logarithmic generalization of Shannon entropy, namely Tsallis entropy, which is non-extensive. Sati and Gupta proposed cumulative residual information based on this non-extensive entropy measure, namely cumulative residual Tsallis entropy (CRTE), and its dynamic version, namely dynamic cumulative residual Tsallis entropy (DCRTE). In the present paper, we propose non-parametric kernel type estimators for CRTE and DCRTE where the considered observations exhibit an -mixing dependence condition. Asymptotic properties of the estimators were established under suitable regularity conditions. A numerical evaluation of the proposed estimator is exhibited and a Monte Carlo simulation study was carried out.

Highlights

Shannon [1] made his signature in statistics by introducing the concept of entropy, a measure of disorder in probability distribution
We propose non-parametric estimators of cumulative residual Tsallis entropy (CRTE) and dynamic cumulative residual Tsallis entropy (DCRTE) using kernel type estimation based on the assumption that underlying lifetimes are assumed to be ρ-mixing
Non-parametric kernel type estimators for CRTE and DCRTE were proproposed for observations which exhibit ρ-mixing dependence

Summary

Introduction

Shannon [1] made his signature in statistics by introducing the concept of entropy, a measure of disorder in probability distribution. Associated with an absolutely continuous random variable X with probability density function (pdf) f ( x ), cumulative distribution function (cdf) F ( x ) and survival function (sf) F ( x ) = 1 − F ( x ), Shannon entropy is defined as Accepted: 19 December 2021 ζ (X) = −. Z where log(·) is the natural logarithm with standard convention 0 log 0 = 0. Nowadays, this measure has gained a peculiar place in sciences such as physics, chemistry, computer sciences, wavelet analysis, image recognition and fuzzy sets. Following the pioneering work of Shannon, the available literature has generated a significant amount of papers related to it, obtained by incorporating some additional parameters which make these entropies sensitive to different the shapes of probability distributions

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