Consider the spacings between i.i.d. uniform observations in the interval (0,1). We develop a general method for evaluating the distribution of the minimum or the maximum of random variables which are sums of consecutive spacings. We then present some applications of this method. 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 an algorithm for the systematic application of this recursion. Our method can be used to solve a variety of problems involving sums of spacings or exponential random variables. In particular, our method gives another way to obtain the distribution of the scan statistic. Because the output of our procedure is a polynomial whose coefficients are computed exactly, we can supply numerical answers which are accurate to any required degree of precision.
Computational Statistics and Data Analysis 26(2), pp.117-132