A New Convexity-Based Inequality, Characterization of Probability Distributions, and Some Free-of-Distribution Tests
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A NEW CONVEXITY-BASED INEQUALITY, CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS, AND SOME FREE-OF-DISTRIBUTION TESTS I. V. Volchenkova∗ and L. B. Klebanov†
UDC 519.2
A goal of the paper is to prove new inequalities connecting some functionals of probability distribution functions. These inequalities are based on the strict convexity of functions used in the definition of the functionals. The starting point is the paper “Cram´er–von Mises distance: probabilistic interpretation, confidence intervals and neighborhood of model validation” by Ludwig Baringhaus and Norbert Henze. The present paper provides a generalization of inequality obtained in probabilistic interpretation of the Cram´er–von Mises distance. If the equality holds there, then a chance to give characterization of some probability distribution functions appears. Considering this fact and a special character of the functional, it is possible to create a class of free-of-distribution two sample tests. Bibliography: 3 titles.
1. Introduction The starting point of the present research is the paper by Ludwig Baringhouse and Norbert Henze [1]. The first result of this paper that interested us is a probabilistic interpretation of the Cramer–von Mises test. We are going to generalize this result, as well as to find tests that do not depend on the type of law, the goodness-of-fit with which we check (statistical test which are free of distribution), and some characterizations of probability distributions. 2. The main results in the one-dimensional case Our first result can be formulated as follows. Theorem 2.1. Let h be a strictly convex continuously differentiable function on the interval [0, 1], such that h(0) = 0. Assume that F (x) and G(x) are continuous distribution functions on the real axis R1 . Then ∞ ∞ 1 h(F (x))dG(x) + h(G(x))dF (x) ≥ 2 h(u)du. (2.1) −∞
−∞
0
Moreover, equality is attained if and only if F (x) = G(x). Remark 2.1. The convexity condition for the function h(x) in Theorem 2.1 can be replaced by the concavity condition, in which case the inequality sign should be replaced by the opposite. The right-hand side of inequality (2.1) allows us to make the following probabilistic interpretation. Namely, let X and Y be random variables with probability distribution functions F (x) and G(x), respectively. Provided that Theorem 2.1 holds, we have 1 E h(F (Y )) + E h(G(X)) ≥ 2
h(u)du. 0
d
Moreover, equality is attained if and only if X = Y , in other words, if X and Y are equally distributed. Note that the statistics representation of the Cramer–von Mises test obtained in [1] is a special case of (2.1) for h(u) = u2 . Nevertheless, a similar interpretation can be used in more general cases. ∗ Czech
Technical University, Prague, Czech Republic.
† Charles
University, Prague, Czech Republic, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 63–76. Original article submitted October 2, 2018. 38 1072-3374/20/2511-0038 ©2020 Springer Science+Business Media, LLC
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