Weighted mean inactivity time function with applications

The concept of mean inactivity time plays a crucial role in reliability, risk theory and life testing. In this regard, we introduce a weighted mean inactivity time function by considering a non-negative weight function. Based on this function, we provide expressions for the variance of transformed random variable and the weighted generalized cumulative entropy. The latter concept is an important measure of uncertainty which is shift-dependent and is of interest in certain applied contexts, such as reliability or mathematical neurobiology. Moreover, based on the comparison of mean inactivity times of a certain function of two lifetime random variables, we introduce and study a new stochastic order in terms of the weighted mean inactivity time function. Several characterizations and preservation properties of the new order under shock models, random maxima and renewal theory are discussed.Comment: 25 page

A NEW GENERALIZED VARENTROPY AND ITS PROPERTIES

The variance of Shannon information related to the random variable $X$, which is called varentropy, is a measurement that indicates, how the information content of $X$ is scattered around its entropy and explains its various applications in information theory, computer sciences, and statistics. In this paper, we introduce a new generalized varentropy based on the Tsallis entropy and also obtain some results and bounds for it. We compare the varentropy with the Tsallis varentropy. Moreover, we explain the Tsallis varentropy of the order statistics and analyse this concept in residual (past) lifetime distributions and then introduce two new classes of distributions by them

Analysis and applications of the residual varentropy of random lifetimes

In reliability theory and survival analysis, the residual entropy is known as a measure suitable to describe the dynamic information content in stochastic systems conditional on survival. Aiming to analyze the variability of such information content, in this paper we introduce the variance of the residual lifetimes, "residual varentropy" in short. After a theoretical investigation of some properties of the residual varentropy, we illustrate certain applications related to the proportional hazards model and the first-passage times of an Ornstein-Uhlenbeck jump-diffusion process