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points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model
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.
[1] M. T. L. Bizley: Derivation of a new formula for the number of minimal lattice paths from $(0, 0)$ to $(km, kn)$ having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuar. 80 (1954), 55-62. DOI 10.1017/S002026810005424X | MR 0061567
[2] M. Dwass: Simple random walk and rank order statistics. Ann. Math. Statist. 38 (1967), 1042-1053. DOI 10.1214/aoms/1177698773 | MR 0215463 | Zbl 0162.50204
[3] N. L. Geller: Two limiting distributions for two-sample Kolmogorov-Smirnov type statistics. Report No. 5, Centre for System Science, Case Western Reserve University, Cleveland, Ohio, (1971).
[4] I. Očka: Simple random walk and rank order statistics. Apl. mat. 22 (1977), 272-290. MR 0438583
[5] K. Sarkadi: On Galton's rank order test. Publ. Math. Inst. Hungar. Acad. Sci. 6 (1961), 125-128.
[6] Z. Šidák: Applications of Random Walks in Nonparametric Statistics. Bull. Internat. Statist. Inst., Proc. of the 39th session, vol. 45 (1973), book 3, 34-42. MR 0356357
[7] G. P. Steck: The Smirnov two-sample tests as rank tests. Ann. Math. Statist. 40 (1969), 1449-1466. DOI 10.1214/aoms/1177697516 | MR 0246473 | Zbl 0186.52304
[8] G. P. Steck G. J. Simmons: On the distributions of $R_{mn}^+$ and $(D_{mn}^+, R_{mn}^+)$. Studia Sci. Math. Hung. 8(1973), 79-89. MR 0339374
[9] I. Vincze: Einige Zweidimensionale Verteilungs und Grenzverteilungssätze in der Theorie der geordneten Stichproben. Publ. Math. Inst. Hungar. Acad. Sci. 2 (1957), 183 - 209. MR 0105172
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