Steck, G. P.; Simmons, G. J. On the distributions of R\(^+_{mn}\) and (D\(^+_{mn}\), R\(^+_{mn}\)). (English) Zbl 0277.62011 Stud. Sci. Math. Hung. 8(1973), 79-89 (1974). Let \(F_m\) and \(G_n\) be the empirical distribution functions of two independent samples of independent, identically distributed, continuous random variables of sizes \(m\) and \(n\), respectively. The difference \(F_m(x) - G_n(x)\) changes only at a \(z_i\), \(i = 1,2,\dots,m+n\), corresponding to one of the observations. Let \(R_{mn}^+\) denote the subscript \(i\) for which \(F_m(z_i)-G_n(z_i)\) first achieves its maximum value \(D_{mn}^+\). Let \(p\) denote the greatest common divisor of \(m\) and \(n\) and define \(q= (m+n)/p\). The authors show that the point probabilities for \(R_{mn}^+\) are non-increasing and equal in blocks of length \(q\) and give a formula for their values. In particular, if \(m\) and \(n\) are relatively prime then \(R_{mn}^+\) is uniformly distributed on the integers \(1,2, \dots,m+n\). The authors also give a formula for the point probabilities of the vector \((D_{mn}^+,R_{mn}^+)\). Reviewer: G. P. Steck Page: Show Scanned Page Cited in 2 ReviewsCited in 3 Documents MSC: 62E15 Exact distribution theory in statistics 60C05 Combinatorial probability 62G99 Nonparametric inference PDFBibTeX XMLCite \textit{G. P. Steck} and \textit{G. J. Simmons}, Stud. Sci. Math. Hung. 8, 79--89 (1974; Zbl 0277.62011)