Recently, Alizadeh and Shafaei (2023) introduced some estimators for Varentropy of a continuous random variable. The present article applies these estimators and construct some tests of fit for Inverse Gaussian distribution. Percentage points and type I error of the new tests are obtained and then power values of the proposed tests against various alternatives are computed. The results of a simulation study show that the tests have a good performance in terms of power. Finally, a real data set is used to illustrate the application of the proposed tests.

In this article, we obtain an upper bound for the variance of a function of the residual life random variable for discrete lifetime distributions. As a special case, we find an upper bound for residual Varentropy. Moreover we characterize some discrete distributions by Cauchy-Schwarz inequality. We also get new expressions, bounds and stochastic comparisons involving measures in reliability and information theory.

Given the importance of Varentropy in information theory, and since a closed form cannot be derived for some discrete distributions, we aim to establish bounds for the Varentropy of these distributions and introduce the past Varentropy for discrete random variables. In this article, we first acquired lower and upper bounds for the Varentropy of the Poisson, binomial, negative binomial, and hypergeometric distributions. Since the resulting upper bounds are expressed as squared logarithmic expectations, we provide an equivalent formulation using squared logarithmic difference coefficients. Similarly, we present lower bounds in terms of logarithmic difference coefficients. Furthermore, an upper bound is derived for the variance of a function of discrete reversed residual lifetime function. We also investigate inequalities involving moments of selected functions via the reversed hazard rate and characterize certain discrete distributions by the Cauchy-Schwarz inequality.

In today’s industrial world, effective maintenance plays a key role in reducing costs and improving productivity. This paper introduces goodness-of-fit tests based on information measures, including entropy, extropy, and Varentropy, to evaluate the type of repair in repairable systems. Using system age data after repair, the tests examine the adequacy of the arithmetic reduction of age model of order 1. The power of the proposed tests is compared with classical tests based on martingale residuals and the probability integral transform. Simulation results show that the proposed tests perform better in identifying imperfect repair models. Their application to real data on vehicle failures also indicates that this model provides a good fit.