Kernel estimation of the past entropy function with dependent data

Kernel estimation of past entropy function

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

The past entropy function, introduced by Di Crescenzo and Longobardi
(2002), is viewed as a dynamic measure of uncertainty in past life. This measure find applications in modeling life time data. In the present work we provide non-parametric kernel type estimators for the past entropy function based on complete and censored data. Asymptotic properties of the estimators are established under suitable regularity conditions. Monte-Carlo simulation studies are carried out to compare the performance of the estimators using the meansquared error. The methods are illustrated using real data sets.