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| Titel: |
Evaluation of Some Steck Determinants with Applications |
| Auteur: |
Steck, G. P. |
| Verschenen in: |
Communications in statistics |
| Paginering: | Jaargang 3 (1974) nr. 2 pagina's 121-138 |
| Jaar: |
1974 |
| Inhoud: |
Let X1 ≤ X2 ≤ … ≤ Xm and Y1 ≤ Y2 ≤ … ≤ Yn be the order statistics from two independent samples of iid random variables with common continuous distribution function F. Let Z1 ≤ Z2 ≤ … ≤ Zm+n denote the ordered combined sample and let Ri be the rank of Xi among the Z's. Steck (1969) proved that for non-decreasing sequences of integers {bi} and {ci} [image omitted] where [image omitted] . The proof was overly complicated and later Mohanty (1971), Pitman (1972) and Sarkadi (1973) gave simpler proofs. In this paper we evaluate certain of these determinants assuming 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 Rvaceva (1952) for the distributions of the one and two-sided Smirnov statistics. The Prinicipal 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 theory of Lagrange series. |
| Uitgever: |
Taylor & Francis |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
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