Evaluation of some steck determinants with applications
George P. Steck · Communications in Statistics · 1974
Let and be the order statistics from two independent samples of iid random variables with common continuous distribution function F. Let denote the ordered combined sample and let Ribe the rank of Xiamong the Z's. Steck (1969) proved that for non-decreasing sequences of integers {b1}and {ci} where The proof was overly complicated and later Mohanty (1971), Pitman (1972) and Sarkadi (1973) gave simpler proofs. In this paper wer evaluate certain of these determd nants assumding n = mb and use them to obtain expressions for the distribution of the two-sided Smirnov statistic and the doubly truncated one-sided Smirnov statistic. In the process we also rederive the combinatorial expressions found by Korolyuk (1955) and Gnedenko and Rvačeva (1952) for the distributions of the one and two-sided Smirnov statistics The principal technique for evaluating these determinants is to modify them by repeated use of Vandermonde type convolution formulae until they become recognizable as a determinant whose value is known. In evaluating the two-sided Smirnov distribution we shall also have occasion to use some results from the theroy of Lagrange series.