ON DATA DRIVEN NEYMAN’S TESTS
Wilbert C. M. Kallenberg
Teresa Ledwina
Abstract: The smooth tests for testing uniformity were introduced by Neyman [15]. The data
driven method of selecting the number of components in a smooth test for uniformity is
discussed, including the first-order asymptotic null distribution, consistency, empirical critical
values and Monte Carlo powers. The first-order asymptotic null distribution is not sufficiently
precise for approximation tools. A substantial improvement is made in this paper by deriving
a second-order approximation of the null distribution, which turns out to be very accurate in
numerical examples. The approximations are based on the second-order behaviour of
Schwarz’s selection rule under uniformity. The new results on are of independent
interest.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -