Large deviations of U-empirical Kolmogorov–Smirnov tests and their efficiency

Abstract

Non-degenerate U -empirical Kolmogorov-Smirnov tests are studied and their large deviation asymptotics under the null-hypothesis is described. Several examples of such statistics used for testing goodness-of-fit and symmetry are considered. It is shown how to calculate their local Bahadur efficiency.

Tests of exponentiality based on Arnold–Villasenor characterization and their efficiencies

“…The family of kernels {Ξ 3 (X, Y, Z; t)}, t ≥ 0, is centered and bounded in the sense given in [26]. Applying the large deviation theorem for the supremum of the family of non-degenerate U-and V -statistics from [26], we get the following result.…”

“…Hence our family of kernels Ξ 3 (X, Y, Z; t) is non-degenerate in the sense described in [26] and δ Using the same reasoning as in the case D…”

“…The kernel Ψ 2 is centered, non-degenerate and bounded. Applying the results on large deviations of non-degenerate U-and V -statistics from [28] (see also [11], [26]), we state the following theorem:…”

Mathematical Statistics and Limit Theorems

“…The examples of such goodness-of-fit tests together with their asymptotic analysis and related calculation of efficiencies can be found in Baringhaus and Henze (1992), Henze and Meintanis (2002), Morris and Szynal (2001), Muliere and Nikitin (2002), Nikitin (1996b), Nikitin (2010, Nikitin and Volkova (2010), and some other related papers.…”

“…They also proposed suitable Kolmogorov-type and omega-square type tests of symmetry. Some efficiency calculations were then performed in Nikitin (1996a), see also Nikitin (2010).…”