Abstract. We propose two families of scale-free exponentiality tests based on the recent characterization of exponentiality by Arnold and Villasenor. The test statistics are based on suitable functionals of U -empirical distribution functions. The family of integral statistics can be reduced to V -or U -statistics with relatively simple non-degenerate kernels. They are asymptotically normal and have reasonably high local Bahadur efficiency under common alternatives.This efficiency is compared with simulated powers of new tests. On the other hand, the Kolmogorov type tests demonstrate very low local Bahadur efficiency and rather moderate power for common alternatives, and can hardly be recommended to practitioners. We also explore the conditions of local asymptotic optimality of new tests and describe for both families special "most favorable" alternatives for which the tests are fully efficient.
Two new symmetry tests, of integral and Kolmogorov type, based on the characterization by squares of linear statistics are proposed. The test statistics are related to the family of degenerate U-statistics. Their asymptotic properties are explored. The maximal eigenvalue, needed for the derivation of their logarithmic tail behavior, was calculated or approximated using techniques from the theory of linear operators and the perturbation theory. The quality of the tests is assessed using the approximate Bahadur efficiency as well as the simulated powers. The tests are shown to be comparable with some recent and classical tests of symmetry. keywords: symmetry tests, characterization, degenerate U-statistics, second order Gaussian chaos process, approximation of maximal eigenvalue, asymptotic efficiency MSC(2010): 62G10, 62G20, 45C05, 47A58 * bozin@matf.bg.ac.rs † bojana@matf.bg.ac.rs ‡ yanikit47@mail.ru § marcone@matf.bg.ac.rs and described in classical literature, as are some more sophisticated signed rank statistics (see e.g. [27], [11], [16], [5]). Another class contains symmetry tests based on the empirical d.f.'s. Many examples, including the Kolmogorov-Smirnov-and ω 2 -type tests, are described in [20]. This monograph offers an extensive review of various symmetry tests, together with the calculation of their efficiencies.In recent times, introducing tests based on characterizations became a popular direction in goodness-of-fit testing. Such tests are attractive because they employ some intrinsic properties of the probability laws related to the characterization, and therefore they can exhibit high efficiency and power.The first to introduce such symmetry tests were Baringhaus and Henze in [4]. They proposed suitable U-empirical Kolmogorov-Smirnov-and ω 2 -type tests of symmetry based on their characterization. The calculation of Bahadur efficiencies, for the Kolmogorov-type test, was then performed in [21], see also [22]. An integral-type symmetry test, based on the same characterization, was proposed and analyzed by Litvinova in [18].Recently, Nikitin and Ahsanullah [23] built new tests of symmetry with respect to zero, based on the characterization by Ahsanullah [1]. This characterization was generalized and used for construction of similar symmetry tests by Milošević and Obradović [19]. The quality of all these tests was examined using the Bahadur efficiency, which is applicable to the case of non-normal limiting distributions.Here we consider the characterization obtained independently by Wesolowski [28, Corollary 1], and Donati-Martin, Song and Yor [6, Lemma 1]. They proved the following proposition:Let X and Y be i.i.d. random variables such that (X − Y ) 2 and (X + Y ) 2 are equidistributed. Then X and Y are symmetric with respect to zero.Our aim is to build the integral-and the Kolmogorov-type U-empirical tests of symmetry based on this characterization; to explore their asymptotic properties; and to assess their quality via the approximate Bahadur efficiency and the simulated powers.Our test statisti...
In this paper, a new class of goodness of fit tests for exponential distribution is proposed. The tests use the equidistribution characterizations of exponential distribution. Based on the U -empirical Laplace transforms of equidistributed statistics, test statistics of the integral type are formed. They are U -statistics with estimated parameters. Their asymptotic properties are derived. Two families of exponentiality tests from this class, based on two selected characterizations, are presented. The approximate Bahadur efficiency is used to assess their quality. Finally, their simulated powers are calculated and the tests are compared with different exponentiality tests.
Two new tests for exponentiality, of integral-and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests.
In this paper we present a new characterization of Pareto distribution and consider goodness of fit tests based on it. We provide an integral and Kolmogorov-Smirnov type statistics based on U-statistics and we calculate Bahadur efficiency for various alternatives. We find locally optimal alternatives for those tests. For small sample sizes we compare the power of those tests with some common goodness of fit tests.
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