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In 1961, Renyi introduced the concept of parametric family of entropies as a generalization of the Shannon entropy. A significant application of this measure can be found in ecology, statistics and various other areas. Later, a divergence measure based on the Renyi entropy was introduced. In this study, we propose a weighted (shift-dependent) version of the Renyi divergence for two doubly truncated nonnegative random variables. Several bounds are obtained. The effect of monotone transformations on the proposed measure is discussed. Finally, a numerical study is performed to provide estimate of the proposed measure. © Springer Nature Switzerland AG 2020.

Department of Mathematics

School of Advanced Sciences

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Studies in Computational Intelligence

A study on weighted doubly truncated renyi divergence

Doubly truncated data are sometimes encountered in several applications, mainly in survival and astronomical data analysis. This occurs when the data falls between two points. In this study, we focus on the Shannon entropy measure of doubly truncated random variables. We propose ordering and various aging properties based on this measure. Characterizations of some useful life distributions are obtained. It is showed that under certain condition, the proposed measure determines the distribution function uniquely. Some results on discrete distributions are presented. Finally, applications are given. © 2020 The Authors. Published by Atlantis Press SARL. This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal of Statistical Theory and Applications

Properties of shannon entropy for double truncated random variables and its applications

In the present communication, we introduce quantile-based (dynamic) inaccuracy measures and study their properties. Such measures provide an alternative approach to evaluate inaccuracy contained in the assumed statistical models. There are several models for which quantile functions are available in tractable form, though their distribution functions are not available in explicit form. In such cases, the traditional distribution function approach fails to compute inaccuracy between two random variables. Various examples are provided for illustration purpose. Some bounds are obtained. Effect of monotone transformations and characterizations are provided. © 2019 Cambridge University Press.

Probability in the Engineering and Informational Sciences

Quantile-Based Study of (Dynamic) Inaccuracy Measures

In this communication, we study doubly truncated weighted Kullback–Leibler divergence (KLD) between two nonnegative random variables. The proposed measure is a generalization of the dynamic weighted KLD introduced by Yasaei Sekeh et al. (2013). In reliability theory and survival analysis, it plays a significant role to study several aspects of a system when lifetimes fall in a time interval. It is showed that under some conditions, the proposed measure determines the distribution function uniquely. Further, characterization theorems for various lifetime distributions are proved. The effect of the monotone transformation on the proposed measure is studied. Some inequalities and bounds in terms of useful measures are obtained and finally, few applications are provided. © 2019, National Statistical Institute. All rights reserved.

Revstat Statistical Journal

On weighted kullback–leibler divergence for doubly truncated random variables

Recently, a shift-independent information measure known as generalized cumulative entropy of order n (GCE n ) was proposed by Kayal (2016). In this communication, we propose a shift-dependent version of GCE n . Various properties including the effect of transformations, bounds etc. have been discussed. Several relationships of the shift-dependent GCE n with some well-known reliability measures are studied. Few characterization results are obtained. We derive an estimator for the proposed measure via empirical distribution function approach. Large sample properties of the estimator are studied when independent observations are taken from a Weibull distribution. © 2018, © 2018 Taylor & Francis Group, LLC.

Communications in Statistics: Simulation and Computation

A shift-dependent generalized cumulative entropy of order n

Generalized cumulative residual entropy has been shown an alternative uncertainty measure to cumulative residual entropy in the literature. We find its applications in the theory of communication, actuarial and computer science. In this chapter, we consider a shift-dependent version of the generalized cumulative residual entropy for the k-th upper record and its dynamic version. The advantage of the proposed measure over the existing measure is discussed. Various results including some useful properties, effect of affine transformations, bounds, stochastic ordering and aging properties are obtained. In addition, inequalities based on the proportional hazard rate model are derived. Further, we obtain some characterization results for various probability models based on the proposed information measure. A nonparametric estimator of the proposed measure is provided and then, its asymptotic normality is established. © 2019, Springer International Publishing AG, part of Springer Nature.

Lecture Notes on Data Engineering and Communications Technologies

On Weighted Extended Cumulative Residual Entropy of k-th Upper Record

Doubly truncated data appear in some applications with survival and astrological data. Analogous to the doubly truncated discrimination measure defined by Misagh and Yari (2012), a generalized discrimination measure between two doubly truncated non-negative random variables is proposed. Several bounds are obtained. It is remarked that the proposed measure can never be equal to a nonzero constant which is independent of the left and right truncated points. The effect of monotone transformations on the proposed measure is discussed. Finally, a simulation study is added to provide the estimates of the proposed discrimination measure. © 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.