On a characterization of monotone likelihood ratio experiments

Summary Pfanzagl (1962,Zeit. Wahrscheinlichkeitsth.,1, 109–115) showed that a dominated family of probability measures has monotone likelihood ratios with respect to some real valued statistic if there exists a set of tests which has certain nice properties. A similar characterization was given by Dettweiler (1978,Metrika,25, 247–254), who did not assume domination. However, Pfanzagl's result is not a special case of the one proved by Dettweiler. We present a theorem which comprises the results of both authors. Our proof shows that not all conditions introduced by them are needed. Furthermore, we investigate the question concerning the generality we get if we do not assume domination.