# Asymptotic generalized bivariate extreme with random index

## TL;DR

This paper characterizes the limit distributions of bivariate extreme generalized order statistics with random sample sizes, providing conditions for their convergence and illustrating the results with examples.

## Contribution

It fully characterizes the limit distributions of bivariate extreme GOS with random sample sizes and establishes conditions for their weak convergence.

## Key findings

- Characterization of limit distribution functions of bivariate extreme GOS with random sizes
- Necessary and sufficient conditions for weak convergence when sample size is independent
- Sufficient conditions and forms of limit distributions when size and variables are dependent

## Abstract

In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be a random variable (rv). The class of limit distribution functions (df's) of the random bivariate extreme generalized order statistics (GOS) from independent and identically distributed RV's are fully characterized. When the random sample size is assumed to be independent of the basic variables and its df is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random bivariate extreme GOS are obtained. Furthermore, when the interrelation of the random size and the basic rv's is not restricted, sufficient conditions for the convergence and the forms of the limit df's are deduced. Illustrative examples are given which lend further support to our theoretical results.