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We consider a class of testing problems where the null space is the union of  k -1 subgraphs of the form  h   j  ( θ   j  )≤ θ   k  , with  j =1,…, k -1, ( θ  1 ,…, θ   k  ) the unknown parameter, and  h   j   given increasing functions. The data consist of  k  independent samples, assumed to be drawn from a distribution with parameter  θ   j  ,  j =1,…, k , respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/ √n  rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for  k =2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.

The likelihood ratio test for non-standard hypotheses near the boundary of the null – with application to the assessment of non-inferiority