Likelihood based inference for monotone response models. (English) Zbl 1133.62328

Banerjee, Moulinath

Ann. Stat. 35, No. 3, 931-956 (2007).

Summary: 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.

MSC:

62G20	Asymptotic properties of nonparametric inference
62G08	Nonparametric regression and quantile regression
62G05	Nonparametric estimation
62E20	Asymptotic distribution theory in statistics

Keywords:

greatest convex minorant; ICM; likelihood ratio statistic; monotone function; monotone response model; self-induced characterization; two-sided Brownian motion; universal limit

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