ON THE DISTRIBUTION OF THE MAXIMUM AND MINIMUM OF RATIOS OF ORDER STATISTICS.
Abstract
Let X sub i(i = 0,1,...,p) be (p + 1) independent and identically distributed nonnegative random variables each representing the jth order statistic in a random sample of size n from a continuous distribution G(x) of a nonnegative random variable. Let H sub (j,n) (x) be the cumulative distribution function of X sub i(i = 0,1,...,p). Consider the ratios Y sub i = X sub i/X sub o (i = 1,2,...,p). The random variables Y sub i (i = 1,2,...,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 (1967). 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 to the power (-x/theta)), x > 0, theta > 0. Section 2 deals with the distribution of the maximum while Section 3 discusses the distribution of the minimum. Section 4 describes the tables of the percentage points of the two statistics. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1967
- Accession Number
- AD0664475
Entities
People
- R. E. Barlow
- S. Panchapakesan
- Sumedha Gupta
Organizations
- Purdue University