Nonstandard likelihood ratio test in exponential families.

Let (pθ)θ∈Θ be an exponential family in ℝk. After establishing nonstandard results about large deviations of the sample mean $$ \overline X $$, this paper defines the nonstandard likelihood ratio test of the null hypothesis H0 : θ ∈ hal($$ \widetilde\Theta _0 $$), where $$ \widetilde\Theta _0 $$ is a standard subset of Θ and hal($$ \widetilde\Theta _0 $$) its halo. If α is the level of the test, depending on whether lnα/n is infinitesimal or not we obtain different rejection criteria. We calculate risks of the first and second kinds (external probabilities) and prove that this test is more powerful than any "regular" nonstandard test based on $$ \overline X $$. [ABSTRACT FROM AUTHOR]