Two-dimensional Kolmogorov-type goodness-of-fit tests based on characterisations and their asymptotic efficiencies

Abstract

In this paper, new two-dimensional goodness-of-fit tests are proposed. They are of supremum type and are based on two different types of characterisations. The first type are those that involve functional equations that the distribution function satisfies, while the second type uses independence of some statistics. The asymptotics of the statistics is studied and Bahadur efficiencies of the tests against some close alternatives are calculated. In the process, a theorem on large deviations of Kolmogorov-type statistics has been extended to the multidimensional case.