Spacings-ratio empirical processes
The paper under review is to consider an empirical process based on ratios of selected pairs of spacings generated by independent samples of arbitrary sizes, and to show that this empirical process converges to a mean centered Brownian bridge provided both samples are uniformly distributed on intervals of equal lengths. Let \(X=\{X_1, \cdots, X_{n_1}\}\) and \(Y=\{Y_1, \cdots, Y_{n_2}\}\) be two independent samples of sizes \(n_1\) and \(n_2\), composed of i.i.d.\ random variables generated by the continuous distribution function \(F_X\) and \(F_Y\). A motivation of this paper under review is to construct nonparametric tests of the null hypothesis (H.0) against the alternative (H.1). \noindent (H.0): There exists a \(\mu \in {\mathbb R}\) such that \(F_X(t) = F_Y(t-\mu)\) for all \(t \in {\mathbb R}\). \noindent (H.1): There does not exist any \(\mu \in {\mathbb R}\) such that \(F_X(t) = F_Y(t-\mu)\) for all \(t \in {\mathbb R}\). The Wilcoxon test, the Mann-Whitney test and the Hodges-Lehmann shift estimator are used to estimate the shift value \(\mu\) when (H.0) holds or to test the assumption that \(\mu =0\) against the alternative that \(\mu\in {\mathbb R}\) is arbitrary. There is no general distribution-free nonparametric methodology to test (H.0) against a general alternative. A main result Theorem 1.1 is that under (H.0) and the distribution of the sample components are uniform on intervals of \({\mathbb R}\), \(\{\gamma_n (u): 0\leq u \leq 1\}\) can be approximated by mean centered Brownian bridges \(\{B(u)-12u(1-u)\int_0^1B(s)ds: 0\leq u \leq 1\}\), where \(\{B(u): 0\leq u \leq 1\}\) is a Brownian bridge. The limiting process of \(\gamma_n (\cdot)\) under (H.0) depends upon \(F(\cdot)\). The main arguments of Theorem 1.1 are presented in Section 3, according to the study of the empirical processes based on sums and ratios of independent exponentially distributed random variables. In Section 4 the authors give the construction of the Gaussian processes for approximations of the empirical processes. The proof of Theorem 1.1 is completed in Section 5. An extension of Theorem 1.1 is given in Theorem 1.2 with more precise rate of convergency.
scientific article; zbMATH DE number 5990588
scientific article