Likelihood Ratio Testing under Measurement Errors

We consider the likelihood ratio test of a simple null hypothesis (with density [Formula: see text]) against a simple alternative hypothesis (with density [Formula: see text]) in the situation that observations [Formula: see text] are mismeasured due to the presence of measurement errors. Thus instead of [Formula: see text] for [Formula: see text] we observe [Formula: see text] with unobservable parameter [Formula: see text] and unobservable random variable [Formula: see text]. When we ignore the presence of measurement errors and perform the original test, the probability of type I error becomes different from the nominal value, but the test is still the most powerful among all tests on the modified level. Further, we derive the minimax test of some families of misspecified hypotheses and alternatives. The test exploits the concept of pseudo-capacities elaborated by Huber and Strassen (1973) and Buja (1986). A numerical experiment illustrates the principles and performance of the novel test.