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 H₀ : ϑ₁ = ϑ₂ versus H_A : 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.