Hodges-Lehmann optimality of tests

Abstract

At several places in the literature there are indications that many tests are optimal in the sense of Hodges-Lehmann efficiency. It is argued here that shrinkage of the acceptance regions of the tests to the null set in a coarse way is already enough to ensure optimality.This type of argument can be used to show optimality of e.g. Kolmogorov-Smirnov tests, Cram&-von Mises tests, and likelihood ratio tests and many other tests in exponential families.

Hodges–Lehmann optimality for testing moment conditions

“…There are results in the literature which indicate that several tests can be Hodges-Lehmann optimal in standard testing problems, such as parameter hypothesis and goodness of fit testing problems (see, Kallenberg and Kourouklis, 1992;Tusnády, 1977). In particular, Kallenberg and Kourouklis (1992) show that the Hodges-Lehmann optimality emerges in general when the acceptance region of a test converges to the set of measures for the null hypothesis in a coarse way, provided the mapping T is continuous in the τ -topology.…”

“…In particular, Kallenberg and Kourouklis (1992) show that the Hodges-Lehmann optimality emerges in general when the acceptance region of a test converges to the set of measures for the null hypothesis in a coarse way, provided the mapping T is continuous in the τ -topology. We show that their continuity assumption in the τ -topology can be replaced with a lower semicontinuity assumption or its localized version in the weak topology.…”

“…Second, we provide novel sets of sufficient conditions for the Hodges-Lehmann optimality. One set is similar to the conditions in Kallenberg and Kourouklis (1992), although we require lower semicontinuity in the weak topology instead of continuity in the τ -topology. The other set involves a localized version of semicontinuity and turns out to be very useful to analyze discontinuous cases.…”

“…These new conditions extend the scope of the Hodges-Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature (e.g. Kallenberg and Kourouklis, 1992). This point is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions (or generalized estimating equations).…”

“…Our Hodges-Lehmann optimality analysis contributes to the literature in several ways. First, we show that the existing general sufficient conditions by Kallenberg and Kourouklis (1992) for a test to be Hodges-Lehmann optimal are too strong for our applications of interest. We provide an example where the empirical likelihood test does not satisfy an even weaker version of those sufficient conditions.…”