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July 2007 Likelihood based inference for monotone response models
Moulinath Banerjee
Ann. Statist. 35(3): 931-956 (July 2007). DOI: 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 n1/3 (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 χ2 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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Moulinath Banerjee. "Likelihood based inference for monotone response models." Ann. Statist. 35 (3) 931 - 956, July 2007.


Published: July 2007
First available in Project Euclid: 24 July 2007

zbMATH: 1133.62328
MathSciNet: MR2341693
Digital Object Identifier: 10.1214/009053606000001578

Primary: 62E20 , 62G08 , 62G20

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

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • July 2007
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