The basic motivation and primary goal of this paper is a qualitative evaluation of the performance of a new weighted statistic for a nonparametric test for stochastic dominance based on two samples, which was introduced in Ledwina and Wy lupek (2012a). For this purpose, we elaborate a useful variant of Kallenberg's notion of intermediate efficiency. This variant is general enough to be applicable to other nonparametric problems. We provide a formal definition of the proposed variant of intermediate efficiency, describe the technical tools used in its calculation, and provide proofs of related asymptotic results. Next, we apply this approach to calculating the intermediate efficiency of the new test with respect to the classical one-sided Kolmogorov-Smirnov test, which is a recognized standard for this problem. It turns out that for a very large class of convergent alternatives the new test is more efficient than the classical one. We also report the results of an extensive simulation study on the powers of the tests considered, which shows that the new variant of intermediate efficiency reflects the exact behavior of the power well.Mathematics Subject Classification. 62G10, 62G20, 60E15.

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In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (Dziedziul K., Kucharska M., Wolnik B., Estimation of the smoothness parameter ). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density functions.

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A class of discrete probability distributions contains distributions with limited support, i.e. possible argument values are limited to a set of numbers (typically consecutive). Examples of such data are results from subjective experiments utilizing the Absolute Category Rating (ACR) technique, where possible answersAn interesting subclass of those distributions are distributions limited to two parameters: describing the mean value and the spread of the answers, and having no more than one change in the probability monotonicity. In this paper we propose a general distribution passing those limitations called Generalized Score Distribution (GSD). The proposed GSD covers all spreads of the answers, from very small, given by the Bernoulli distribution, to the maximum given by a Beta Binomial distribution. We also show that GSD correctly describes subjective experiments scores from video quality evaluations with probability of 99.7%. A Google Collaboratory website with implementation of the GSD estimation, simulation, and visualization is provided.