June, 1969 On the Distribution of the Maximum and Minimum of Ratios of Order Statistics
R. E. Barlow, S. S. Gupta, S. Panchapakesan
Ann. Math. Statist. 40(3): 918-934 (June, 1969). DOI: 10.1214/aoms/1177697597

## Abstract

Let $X_i(i = 0, 1, \cdots, p)$ be $(p + 1)$ independent and identically distributed nonnegative random variables each representing the $j$th order statistic in a random sample of size $n$ from a continuous distribution $G(x)$ of a nonnegative random variable. Let $G_{j,n}(x)$ be the cumulative distribution function of $X_i(i = 0, 1, \cdots, p)$. Consider the ratios $Y_i = X_i/X_0 (i = 1, 2, \cdots, p)$. The random variables $Y_i (i = 1, 2, \cdots, p)$ are correlated and the distribution of the maximum and the minimum is of interest in problems of selection and ranking for restricted families of distribution. The distribution-free subset selection rules using the percentage points of these order statistics are investigated in a companion paper by Barlow and Gupta (1969). In the present paper, we discuss the distribution of these statistics, in general, for any $G(x)$ and then derive specific results for $G(x) = 1 - e^{-x/\theta}, x > 0, \theta > 0$. Section 2 deals with the distribution of the maximum while Section 3 discusses the distribution of the minimum. Some asymptotic results are given in Section 4, while Section 5 describes the tables of the percentage points of the two statistics.

## Citation

R. E. Barlow. S. S. Gupta. S. Panchapakesan. "On the Distribution of the Maximum and Minimum of Ratios of Order Statistics." Ann. Math. Statist. 40 (3) 918 - 934, June, 1969. https://doi.org/10.1214/aoms/1177697597

## Information

Published: June, 1969
First available in Project Euclid: 27 April 2007

zbMATH: 0181.45903
MathSciNet: MR267709
Digital Object Identifier: 10.1214/aoms/1177697597