Locally asymptotically optimal tests in semiparametric generalized linear models in the 2-sample-problem

Summury Let ( X i,j , Y i,j ), i = 1,…, n , j = 1,2, be a sample from two populations, where the X i,j are d -dimensional covariates which have an effect on the response variable Y i,j . It is assumed that the conditional distribution of Y i,j given X i,j = x is Q g(αj + βj T x ) where { Q ϑ | ϑ ∊ Θ}, Θ ⊆ R, is a parent family, g is the so-called link function and ϑ j = (α j ,β j ) the parameters of interest. Using the LAN theory, a sequence of locally asymptotically optimal tests φ ^ n for H 0 : ϑ 1 = ϑ 2 versus H A : ϑ 1 ≠ ϑ 2 is constructed for an unknown link function g . These tests are asymptotic maximin-tests and adaptive in the sense that the plugging-in of an estimator for the nuisance parameters g does not reduce the local asymptotic power compared to the situation of a known nuisance parameter g . To attain exact α-tests even for finite sample size a permutation test version is given with the same local asymptotic power as φ ^ n .