Fixed-αand fixed-βefficiencies

Christopher S. Withers, Saralees Nadarajah
2013 E S A I M: Probability & Statistics  
Consider testing H0 : F ∈ ω0 against H1 : F ∈ ω1 for a random sample X1, . . . , Xn from F , where ω0 and ω1 are two disjoint sets of cdfs on R = (−∞, ∞). Two non-local types of efficiencies, referred to as the fixed-α and fixed-β efficiencies, are introduced for this two-hypothesis testing situation. Theoretical tools are developed to evaluate these efficiencies for some of the most usual goodness of fit tests (including the Kolmogorov-Smirnov tests). Numerical comparisons are provided using
more » ... re provided using several examples. Mathematics Subject Classification Article published by EDP Sciences c EDP Sciences, SMAI 2013 FIXED-α AND FIXED-β EFFICIENCIES 225 and We wish to test whether H 0 : F ∈ ω 0 against H 1 : F ∈ ω 1 , where ω 0 and ω 1 are two disjoint sets of cdfs on R. For example, ω 0 = {F 0 } and ω 1 = {F | F = F 0 }. As candidates for the test, we consider the class of statistics given by (1.1)-(1.7). This class consists of the integral, Kolmogorov-Smirnov and Kuiper statistics, T n2 (1), T n∞ (1) and V n (1) whose asymptotic null distributions are given in Anderson and Darling [2], Kolmogorov [15] and Stephens [21] 3 . This paper is related to Withers and Nadarajah [23], where we showed how the asymptotic power (AP) of T n2 (ψ) may be computed. Withers and Nadarajah [23] also compared the AP of T n2 (ψ) with the AP of T n2 (1), D n (1), V n (1) for the envelope power function of a particular example, the double-exponential shift family. This paper deals with exact non-local types of efficiencies for the general two-hypothesis testing situation. There are generally three different strategies to try and approximate such efficiencies: taking alternatives close to the null hypothesis leads to Pitman efficiency; small levels are related to Bahadur efficiency [3]; consideration of high powers results in the Hodges-Lehmann [11] efficiency. There are also other strategies due to Chernoff, Kallenberg, Borovkov and Mogulskiy. Hodges-Lehmann and Bahadur efficiencies for comparing the performance of gof tests are very much related to large deviation results. Pitman's efficiency is more connected to the notion of contiguity and is nicely studied in the framework given by Le Cam's theory of statistical experiments. However, Pitman and Hodges-Lehmann efficiencies are not appropriate when test statistics have non-normal limiting distributions, for example, Cramer-von Mises and Watson statistics have degenerate kernels with nonnormal limiting distributions. Furthermore, Hodges-Lehmann efficiency cannot discriminate between two-sided tests like Kolmogorov and Cramer-von Mises tests that are asymptotically optimal. Bahadur efficiencies are not easy to compute. Besides, approximate Bahadur efficiencies are of "little value as measures of performance of tests since monotone transformations of a test statistic may lead to entirely different approximate Bahadur slopes" [14] . So, there is a need for variations of these efficiencies. In this paper, we introduce two new efficiencies that are "intermediate" between the Hodges-Lehmann and Bahadur efficiencies. We provide some tools from the calculus of variations to compute them in some of the most usual nonparametric gof tests: integral and Kolmogorov-Smirnov tests. For a review of results related to this paper, we refer the readers to Wieand [22], Kallenberg and Ledwina [14], Kallenberg and Koning [13], Litvinova and Nikitin [16], and the most excellent book by Nikitin [17] . The contents of this paper are organized as follows. In Section 2, two non-local types of efficiency (e α , e β ) are introduced. These are computed in Sections 3 and 4 for gof tests of the type T nm (ψ) or V n (ψ) for parametric and non-parametric alternatives. It is argued that locally T n∞ (ψ) is preferable to T nm (ψ) if m < ∞, in testing F = F 0 against "F is not close to F 0 ". For α-level tests the Hodges-Lehmann efficiency or its generalization the fixed-α efficiency (Sect. 2) is appropriate, but is shown in Section 3 to tend to one under suitable conditions, for the statistics we consider, when testing F ∈ ω 0 against F ∈ ω 1 as ω 0 shrinks to {F 0 }. Section 4 gives the Bahadur efficiency for some common parametric examples, using large deviation results derived in part from the work of Hoadley [10] and Abrahamson [1]. More interesting is a comparison of the statistics when testing whether F 0 is close to F (sup |F − F 0 | = a 0 , say) or distant from F (sup |F − F 0 | = a 1 , say). This is carried out by computing e β in Section 4 when a 0 = 0, for the statistics T n1 (1), T n2 (1), V n (1) and D n (ψ) for certain ψ. The values of e β for these statistics are compared using several examples: a normal with shift alternative example, a logistic with shift alternative example, a double-exponential with shift alternative example and others. Section 5 establishes a local inefficiency of T nm (ψ). The proofs of all results are given in Section 6.
doi:10.1051/ps/2011143 fatcat:uiya6ee7jnh6ndai3wylit7rf4