Testing for first order stochastic dominance in either direction (Q1966012)

Let \(F\) and \(G\) denote continuous distribution functions and let \(X_1,\dots,X_n\) and \(Y_1,\dots,Y_m\) denote two independent samples from \(F\) and \(G\), respectively. Let \[ \hat{F}_n(x):=\#\{i |X_i\leq x\}/n,\quad \hat{G}_m(x):=\#\{i |Y_i \leq x\}/m \] for \(x \in R\) and \[ D^+_{n,m}:=\sup_{x\in R}(\hat{F}_n(x)-\hat{G}_m(x)),\quad D^-_{n,m}:=\sup_{x\in R}(\hat{G}_n(x)-\hat{F}_m(x)). \] For testing the hypothesis \[ H_o:F(x)\leq G(x)\;\text{ for } x\in R \text{ or } G(x)\leq F(x) \text{ for } x\in R \] \textit{K. Mosler} [see H. Rinne et al. (eds.), Grundlagen der Statistik und ihre Anwendungen, 149-155 (1995; Zbl 0851.62030)] suggested the test statistic \[ \text{Min}=\min\{D^+_{n,m},D^-_{n,m}\}. \] The exact finite sample and asymptotic distribution of the Mosler statistic is derived. Some quantiles are tabulated. The power of the corresponding test is investigated.