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On the distributions of R\(^+_{mn}\) and (D\(^+_{mn}\), R\(^+_{mn}\)). (English) Zbl 0277.62011

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

MSC:

62E15 Exact distribution theory in statistics
60C05 Combinatorial probability
62G99 Nonparametric inference
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