Geodesic
On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$
Applications of Mathematics, Tome 27 (1982) no. 6, pp. 417-425.

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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 $x_i, i=1,2,\ldots, m+n$, corresponding to one of the observations. Let $R^+_{mn}(j)$ denote the subscript $i$ for which $F_m(x_i)-G_n(x_i)$ achieves its maximum value $D^+_{mn}$ for the $j$th time $(j=1,2,\ldots)$. The paper deals with the probabilities for $R^+_{mn}(j)$ and for the vector $(D^+_{mn}, R^+_{mn}(j))$ under $H_0 : F=G$, thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model.
DOI : 10.21136/AM.1982.103988
Classification : 62E15, 62G10, 62G30
Keywords: points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model 
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     title = {On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$},
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Saran, Jagdish; Sen, Kanwar. On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$. Applications of Mathematics, Tome 27 (1982) no. 6, pp. 417-425. doi : 10.21136/AM.1982.103988. http://geodesic.mathdoc.fr/articles/10.21136/AM.1982.103988/

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