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On the distributions of R m n + ( j ) and ( D m n + , R m n + ( j ) )

Jagdish SaranKanwar Sen — 1982

Aplikace matematiky

The contents of the paper is concerned with the two-sample problem where F m ( x ) and G n ( x ) are two empirical distribution functions. The difference F m ( x ) - G n ( x ) changes only at an x i , i = 1 , 2 , ... , m + n , corresponding to one of the observations. Let R m n + ( j ) denote the subscript i for which F m ( x i ) - G n ( x i ) achieves its maximum value D m n + for the j th time ( j = 1 , 2 , ... ) . The paper deals with the probabilities for R m n + ( j ) and for the vector ( D m n + , R m n + ( j ) ) under H 0 : F = G , thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model.

Some distribution results on generalized ballot problems

Jagdish SaranKanwar Sen — 1985

Aplikace matematiky

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes and that all possible a + b a voting sequences are equally probable. Denote by α r and by β r the number of votes registered for A and for B , respectively, among the first r votes recorded, r = 1 , , a + b . The purpose of this paper is to derive, for a b - c , the probability distributions of the random variables defined as the number of subscripts r = 1 , , a + b for which (i) α r = β r - c , (ii) α r = β r - c but α r - 1 = β r - 1 - c ± 1 , (iii) α r = β r - c but α r - 1 = β r - 1 - c ± 1 and α r + 1 = β r + 1 - c ± 1 , where c = 0 , ± 1 , ± 2 , .

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