Likelihood based inference for monotone response models

By Moulinath Banerjee. 2007

60 citations recorded by the source corpus.

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 √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 Χ 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/71a66f7e507fe829cf5c6f73517bf37c1e037ab0

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Citations in the corpus

Divide and conquer in nonstandard problems and the super-efficiency phenomenon (2016, methodology): “, isotonic regression [7, 8], where a monotone regression function is estimated via least squares under that shape constraint, the current status model (and extensions thereof) [5, 16], where the distribution of a failure time is estimated under the monotonicity constraint from discrete response dat”
Nonparametric Shape-Restricted Regression (2017, background): “As shown in [13, 14] (in the setting of random design, which can be easily generalized to cover the uniform grid design; see [7]), if f(t) = φ0 and f ′(t) 6= 0, then Ln d → σ(2)L,”
Local and global asymptotic inference in smoothing spline models (2012, background): “A relevant study was conducted by Banerjee [1], who considered a likelihood ratio test for monotone functions, but his estimation method and null limiting distribution are fundamentally different from ours.”
Generalized additive and index models with shape constraints (2014, background): “Banerjee (2007) and Banerjee (2009) studied monotone regression models in which the conditional distribution of a response given a covariate is assumed to come from a regular parametric model and an exponential family respectively.”