## The Annals of Mathematical Statistics

### On the Distribution of the Maximum and Minimum of Ratios of Order Statistics

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 40, Number 3 (1969), 918-934.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177697597

Digital Object Identifier
doi:10.1214/aoms/1177697597

Mathematical Reviews number (MathSciNet)
MR267709

Zentralblatt MATH identifier
0181.45903

JSTOR