On the asymptotic power of cramér-von mises tests of fit

Summary

X ₁(n),⋯,X _n (n) are independent and identically distributed random variables, the common distribution function known to be absolutely continuous, but otherwise unknown. The problem is to test the hypothesis that the unknown cumulative distribution function is some completely specified cumulative distribution functionF(x). LetH _n (x) denote the empirical cumulative distribution function based onX ₁(n),⋯,X _n (n), and let ø(t) denote a given nonnegative bounded weight function defined for 0≦≦1. In [1] the asymptotic distribution of the Cramer-Von Mises statisticW ² _n , defined as

$$n\int_{ - \infty }^\infty {[F(x) - H_n (x)]^2 \psi (F(x))dF(x),} $$

was found, when the hypothesis being tested is true. For the functions ø(t) considered in [1], this asymptotic distribution is not normal. In this paper it is shown that under alternative cumulative distribution functions not too close toF(x),W ² _n is asymptotically normally distributed.