Chernoff Efficiency and Deficiency

Abstract

In 1952 Chernoff introduced a measure of asymptotic efficiency for tests. Comparison in the sense of Chernoff is concerned with fixed alternatives. In contrast to Bahadur's approach, where the probabilities of first and second kind are treated in an unbalanced way, in Chernoff's approach both probabilities go to zero. For the calculation of Chernoff efficiencies one has to develop large deviation theorems both under the null hypothesis and under the alternative hypothesis. In this paper some basic properties are mentioned and the concept of Chernoff deficiency is introduced in a manner analogous to the Pitman and Bahadur case. It is shown that in typical testing problems in multivariate exponential families, the likelihood ratio test is Chernoff deficient of order $\mathscr{O}(\log n)$. Many of the results agree with corresponding results in the Bahadur case.