Likelihood based inference for monotone response models

Details Likelihood based inference for monotone response models Likelihood based inference for monotone response models

2007-08-16

arxiv.org/abs/0708.2177v1

Annals of Statistics 2007, Vol. 35, No. 3, 931-956

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000001578 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000001578

The behavior of maximum likelihood estimates (MLEs) 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 $n^{1/3}$ (slower than the usual $\sqrt{n}$ rate) with a non-Gaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLEs 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 $\chi^2$ but can be explicitly characterized in terms of a functional of Brownian motion. Applications of the main results are presented and potential extensions discussed.

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