On the construction of least favourable pairs of distributions

Summary Convex programming techniques were used by Witting and Krafft in 4] in order to reduce a testing problem for composite hypotheses to one for simple hypotheses. This is realized in terms of least favourable pairs of distributions, which represent the solution of the dual of a suitable program. Without further assumptions on the hypotheses, however, the results, derived that way (cf. Baumann 1], Österreicher 6] and Kusolitsch and Österreicher 5]), are of less practical impact. This is due to the fact that in this case the least favourable pairs depend on the level of the testing problem. Conditions avoiding this, were given by Huber and Strassen in 3]. These conditions make use of 2-alternating capacities in the sense of Choquet. The present paper offers a rather general principle of constructing the least favourable distribution in the case, when one of the two hypotheses is simple. This method works also for the local variation model and the Prohorov neighbourhood model in the case of monotone likelyhood ratio. For simple cases—subsuming the gross error model and the total variation model, for which the solution was given by Huber in 2]—a least favourable pair is obtained by using the mentioned technique of construction two times successively.