Ordering results for random maxima and minima from two dependent Kumaraswamy generalized distributed samples

TL;DR

This paper investigates stochastic ordering of maxima and minima from two dependent, heterogeneous samples modeled with Kumaraswamy generalized distributions, considering random sample sizes and dependencies.

Contribution

It establishes new stochastic order results for maxima and minima of dependent Kumaraswamy generalized samples with random sizes, extending existing theories.

Findings

Stochastic orders between maxima and minima are derived.

Results include hazard rate, reversed hazard rate, and likelihood ratio orders.

Applicable to dependent samples with random sizes in Kumaraswamy generalized models.

Abstract

Let $\{X_{1},\ldots,X_{N_1}\}$ and $\{Y_{1},\ldots,Y_{N_2}\}$ be two sequences of interdependent heterogeneous samples, where for $i=1,\ldots,N_{1},$ $X_{i}\sim \text{Kw-G}(x, \alpha_{i}, \gamma_{i};G)$ and for $i=1,\ldots,N_{2},$ $Y_{i}\sim \text{Kw-G}(x, \beta_{i}, \delta_{i};H),$ where $G$ and $H$ are baseline distributions in the Kumaraswamy generalized model and $N_1$ and $N_2$ are two positive integer-valued random variables, independently of $X_{i}'$s and $Y_{i}'$s, respectively. In this article, we establish several stochastic orders such as usual stochastic, hazard rate, reversed hazard rate, dispersive and likelihood ratio orders between the random maxima ($X_{{N_1}:{N_1}}$ and $Y_{{N_2}:{N_2}}$) and the random minima ($X_{{1}:{N_1}}$ and $X_{{1}:{N_2}}$), when the sample sizes are different and random (positive).