Open Access
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


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.


Download 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.


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

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

Rights: Copyright © 1969 Institute of Mathematical Statistics

Vol.40 • No. 3 • June, 1969
Back to Top