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

Summury Let (Xi,j, Yi,j), i = 1,…,n, j = 1,2, be a sample from two populations, where the Xi,j are d-dimensional covariates which have an effect on the response variable Yi,j. It is assumed that the conditional distribution of Yi,j given Xi,j = x is Qg(αj + βjTx) where {Qϑ | ϑ ∊ Θ}, Θ ⊆ R, is a parent family, g is the so-called link function and ϑj = (αj,βj) the parameters of interest. Using the LAN theory, a sequence of locally asymptotically optimal tests φ^n for H0 : ϑ1 = ϑ2 versus HA : ϑ1 ≠ ϑ2 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.