On consistency and inconsistency of nonparametric tests
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For χ^2-tests with increasing number of cells, Cramer-von Mises tests, tests generated L_2- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients we find necessary and sufficient conditions of consistency and inconsistency of sequences of alternatives having a given rate of convergence to hypothesis in L_2-norm. We show that asymptotic of type II error probabilities of sums of alternatives of consistent and inconsistent sequences coincide with the asymptotic for consistent sequence. We find analytic assignment of consistent sequences that do not have inconsistent components. We point out the largest convex sets of functions such that the sequences from these sets do not have inconsistent components. We show that these sets are balls in Besov spaces B^s_2∞.
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