Likelihood Ratio Tests Under Local and Fixed Alternatives in Monotone Function Problems

We focus on a class of non-standard problems involving non-parametric estimation of a monotone function that is characterized by n rate of convergence of the maximum likelihood estimator, non-Gaussian limit distributions and the non-existence of ffiffiffi n p -regular estimators. We have shown elsewhere that under a null hypothesis of the type w(z0) 1⁄4 h0 (w being the monotone function of interest) in non-standard problems of the above kind, the likelihood ratio statistic has a universal limit distribution that is free of the underlying parameters in the model. In this paper, we illustrate its limiting behaviour under local alternatives of the form wn(z), where wn(Æ) and w(Æ) vary in O(n ) neighbourhoods around z0 and wn converges to w at rate n 1/3 in an appropriate metric. Apart from local alternatives, we also consider the behaviour of the likelihood ratio statistic under fixed alternatives and establish the convergence in probability of an appropriately scaled version of the same to a constant involving a Kullback–Leibler distance.