The Annals of Mathematical Statistics

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

R. E. Barlow, S. S. Gupta, and S. Panchapakesan

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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.

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

First available in Project Euclid: 27 April 2007

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Barlow, R. E.; Gupta, S. S.; Panchapakesan, S. On the Distribution of the Maximum and Minimum of Ratios of Order Statistics. Ann. Math. Statist. 40 (1969), no. 3, 918--934. doi:10.1214/aoms/1177697597.

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