Weighted Tail Random Variable: A Novel Framework with Stochastic Properties and Applications

TL;DR

This paper presents a new framework for constructing weighted tail random variables using survival functions and weight functions, with applications in reliability analysis and improved data fitting.

Contribution

It introduces the weighted tail random variable (WTRV) framework, exploring its properties, stochastic orderings, and applications to the Kumaraswamy distribution with real data.

Findings

WTRV framework enhances reliability analysis.

WTRV provides better data fit than original distributions.

The framework is applicable to various continuous distributions.

Abstract

This paper introduces a novel framework to construct the probability density function (PDF) of non-negative continuous random variables. The proposed framework uses two functions: one is the survival function (SF) of a non-negative continuous random variable, and the other is a weight function, which is an increasing and differentiable function satisfying some properties. The resulting random variable is referred to as the weighted tail random variable (WTRV) corresponding to the given random variable and the weight function. We investigate several reliability properties of the WTRV and establish various stochastic orderings between a random variable and its WTRV, as well as between two WTRVs. Using this framework, we construct a WTRV of the Kumaraswamy distribution. We conduct goodness-of-fit tests for two real-world datasets, applied to the Kumaraswamy distribution and its corresponding WTRV. The test results indicate that the WTRV offers a superior fit compared to the Kumaraswamy distribution, which demonstrates the utility of the proposed framework.