On the Converse Theorem in Statistical Hypothesis Testing

Abstract

Simple statistical hypothesis testing is investigated by making use of divergence geometric method. The asymptotic behavior of the minimum value of the second-kind error probability under the constraint that the first-kind error probability is bounded above by exp(rn) is looked for, where r is a given positive number. If r is greater than the divergence of the two probability measures, the so-called converse theorem holds. It is shown that the condition under which the converse theorem holds can be divided into two separate cases by analyzing the geodesic connecting the two probability measures and, as a result, a lucid explanation is given for Han—Kobayashi’s linear function fr(X). © 1993 IEEE.