The limit distribution of weighted $L^2$-goodness-of-fit statistics under fixed alternatives, with applications

Abstract

We present a general result on the limit distribution of weighted one- and two-sample $L^2$-goodness-of-fit test statistics of some hypothesis $H_0$ under fixed alternatives. Applications include an approximation of the power function of such tests, asymptotic confidence intervals of the distance of an underlying distribution with respect to the distributions under $H_0$, and an asymptotic equivalence test that is able to validate certain neighborhoods of $H_0$.