Kernel Estimation of Tsallis Entropy and its Generalization for Length-biased Data

Zamini, Raheleh; Ajami, Masoud; Parvizi, Sepide

doi:10.22034/jirss.2024.2016847.1046

Kernel Estimation of Tsallis Entropy and its Generalization for Length-biased Data

Document Type : Original Article

Authors

Raheleh Zamini ¹

Masoud Ajami ²

Sepide Parvizi ¹

¹ Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.

² Department of Statistics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

10.22034/jirss.2024.2016847.1046

Abstract

A new generalized Shannon entropy is Tsallis entropy. The Shannon entropy is additive and the Tsallis entropy is, however, non-additive. Due to the flexibility of the Tsallis entropy compared to the Shannon entropy, the non-additive entropy measures find their justification in many areas. In this paper, we propose two non-parametric kernel estimators for the Tsallis entropy and two non-parametric kernel estimators for the residual Tsallis entropy for the length-biased data. We investigate some asymptotic properties for these estimators such as the consistency and asymptotic normality. We obtain the bias, variance and the mean integrated squared error (MISE) of estimators. We also compare the behaviour of proposed estimators using the Monte Carlo simulation and plot some figures to see how close the fitted distribution is to the histogram of the data. In the end, we use a real dataset to show the performance of the proposed estimators.

Keywords

Asymptotic normality

Bandwidth

Kernel

Length-biased data

Residual Tsallis entropy

Strong consistency

Tsallis entropy

Subjects

Non-Parametric Statistics

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