Likelihood Based Inference For Monotone Functions: Towards a General Theory

The behavior of maximum likelihood estimates (MLE’s) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate n (slower than the usual √ n rate) with a non-Gaussian limit distribution. A unified framework for likelihood based estimation of monotone functions is developed and very general limit theorems describing the behavior of the MLE’s and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer χ but can be explicitly characterized in terms of a functional of Brownian motion. Special instances of the general theorems and potential extensions are discussed.