Composite Tests under Corrupted Data

Abstract

This paper focuses on test procedures under corrupted data. We assume that the observations Z i are mismeasured, due to the presence of measurement errors. Thus, instead of Z i for i = 1 , … , n, we observe X i = Z i + δ V i, with an unknown parameter δ and an unobservable random variable V i. It is assumed that the random variables Z i are i.i.d., as are the X i and the V i. The test procedure aims at deciding between two simple hyptheses pertaining to the density of the variable Z i, namely f 0 and g 0. In this setting, the density of the V i is supposed to be known. The procedure which we propose aggregates likelihood ratios for a collection of values of δ. A new definition of least-favorable hypotheses for the aggregate family of tests is presented, and a relation with the Kullback-Leibler divergence between the sets f δ δ and g δ δ is presented. Finite-sample lower bounds for the power of these tests are presented, both through analytical inequalities and through simulation under the least-favorable hypotheses. Since no optimality holds for the aggregation of likelihood ratio tests, a similar procedure is proposed, replacing the individual likelihood ratio by some divergence based test statistics. It is shown and discussed that the resulting aggregated test may perform better than the aggregate likelihood ratio procedure.