Likelihood based inference for monotone response models

TL;DR

This paper develops a likelihood-based framework for estimating monotone functions, revealing that MLEs converge at a rate of n^{1/3} with non-Gaussian limits, and characterizes the asymptotic distribution of the likelihood ratio statistic.

Contribution

It introduces a new likelihood-based approach for monotone function estimation, deriving limit theorems and explicit distributions for MLEs and likelihood ratio statistics.

Findings

MLE of monotone functions converges at rate n^{1/3}

Likelihood ratio statistic has a non-chi-squared limit distribution

Explicit characterization of the limit distribution in terms of Brownian motion

Abstract

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.