COMPUTING THE JOINT DISTRIBUTION OF GENERAL LINEAR COMBINATIONS OF SPACINGS OR EXPONENTIAL VARIATES

Abstract

We present an algorithm for computing exact expressions for the distribution of the maximum or minimum of an arbitrary finite collection of linear combinations of spacings or exponential random variables with rational coefficients. These expressions can then be manipulated or evaluated using symbolic math packages such as MAPLE. As examples, we apply this algorithm to obtain the distributions of the maximum and minimum of a moving average process, and the distribution of the Kolmogorov-Smirnov statistic. Key words and phrases: Kolmogorov-Smirnov statistic, moving average process, symbolic computations. 1 Introduction Let S (n) denote the vector of spacings between n random points on the interval (0; 1). More precisely, suppose that X 1 ; X 2 ; : : : ; X n are i.i.d. from a uniform distribution on the interval (0, 1), and let X (1) X (2) \\Delta \\Delta \\Delta X (n) be the corresponding order statistics. We define the spacings S 1 ; S 2 ; : : : ; S n+1 to be the successive di...