Weighted mean inactivity time function with applications

TL;DR

This paper introduces a weighted mean inactivity time function, explores its properties, and applies it to reliability and neurobiology, providing new measures of uncertainty and stochastic ordering.

Contribution

It proposes a novel weighted mean inactivity time function, derives related uncertainty measures, and establishes a new stochastic order with various preservation properties.

Findings

Derived expressions for variance and weighted generalized cumulative entropy.

Introduced a new stochastic order based on weighted mean inactivity times.

Analyzed properties under shock models, maxima, and renewal processes.

Abstract

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.