Abstract
The estimation of the entropy of a random system or process is of interest in many scientific applications. The aim of this article is the analysis of the entropy of the famous Kumaraswamy distribution, an aspect which has not been the subject of particular attention previously as surprising as it may seem. With this in mind, six different entropy measures are considered and expressed analytically via the beta function. A numerical study is performed to discuss the behavior of these measures. Subsequently, we investigate their estimation through a semi-parametric approach combining the obtained expressions and the maximum likelihood estimation approach. Maximum likelihood estimates for the considered entropy measures are thus derived. The convergence properties of these estimates are proved through a simulated data, showing their numerical efficiency. Concrete applications to two real data sets are provided.
Highlights
Information theory provides natural mathematical tools for measuring the uncertainty of random variables and the information shared by them
Reference [9] estimated the entropy of the Weibull distribution by considering different loss functions based on a generalized progressively hybrid censoring scheme
Based on n values x1, . . ., xn supposed to be observed from a random variable X with the Kumaraswamy distribution with parameters a and b, the maximum likelihood estimates (MLEs) of a and b, say a^ and b^, are defined by ða^; b^Þ 1⁄4 argmax ða;bÞ2ð0;þ1Þ2 ‘ða; bÞ; where l(a, b) denotes the log-likelihood function specified by
Summary
Information theory provides natural mathematical tools for measuring the uncertainty of random variables and the information shared by them. Reference [9] estimated the entropy of the Weibull distribution by considering different loss functions based on a generalized progressively hybrid censoring scheme. Reference [18] provided an exact expression for entropy information contained in both types of progressively hybrid censored data and applied it in the setting of the exponential distribution. Reference [20] presented the estimation of entropy for inverse Weibull distribution under multiple censored data. Numerical values of these entropy measures with different values of the parameters are given.
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