The Shortcoming of Locally Most Powerful Tests in Curved Exponential Families

Abstract

Comparison of tests with respect to contiguous alternatives is mostly concerned with fixed levels. Properties of locally most powerful (LMP) tests in this sense are well-known in statistical literature. In this note the behaviour of LMP tests is studied for local (not necessarily contiguous) alternatives and vanishing levels of significance. It turns out that the shortcoming of the LMP test tends to zero at the rate $n^{-1} |\log \alpha_n|^{3/2}$.