Efficient robust tests in parametric models

SummaryLet {Pθ:θ∈Θ}, Θ an open subset of Rk, be a regular parametric model for a sample of n independent, identically distributed observations. Formulated and solved in this paper is a robust version of the classical multi-sided hypothesis testing problem concerning θ, or a subvector of θ. In the robust testing problem, the usual parametric null hypothesis and alternatives are both replaced with larger, more realistic, sets of possible distributions for each observation. These sets, defined in terms of a Hellinger metric projection of the actual distribution onto a subspace associated with the parametric null hypothesis, are required to shrink as sample size increases, so as to avoid trivial asymptotics. One construction of an asymptotically minimax test for the robust testing problem is based upon the robust estimate of θ developed in Beran (1979); another construction amounts to an adaptively modified C(α) test.