Computing the Exact Distribution of the Extremes of Linear Combinations of Spacings

We develop a methodology for evaluating the distribution of the minimum or the maximum of linear combinations of spacings and then present some applications of this methodology. The main idea underlying our approach was given by Huffer (1988): A recursion is used to break up the joint distribution of several linear combinations of spacings into a sum of simpler components. We continue applying this recursion until we obtain components which are simple and easily expressed in closed form. In this paper we propose algorithms for the systematic application of this recursion. Our methods are fairly general and can be used to solve a variety of problems involving linear combinations of spacings or exponential random variables. A special case of great importance is when the linear combinations are simply sums of consecutive spacings. We develop specialized methods for dealing with this case. Because the output of our procedure is a polynomial whose coefficients are computed exactly, we can s..