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December 2010 Nonparametric estimation of multivariate convex-transformed densities
Arseni Seregin, Jon A. Wellner
Ann. Statist. 38(6): 3751-3781 (December 2010). DOI: 10.1214/10-AOS840


We study estimation of multivariate densities p of the form p(x)=h(g(x)) for x∈ℝd and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y)=ey for y∈ℝ;; in this case, the resulting class of densities $$\mathcal {P}(e^{-y})=\{p=\exp(-g): g\mbox{ is convex}\} $$ is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class.

We first investigate when the maximum likelihood estimator exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations h, including decreasing and increasing functions h. The resulting models for increasing transformations h extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to h(y)=exp(y).

We then establish consistency of the maximum likelihood estimator for fairly general functions h, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of p and its vector of derivatives at a fixed point x0 under natural smoothness hypotheses on h and g. The proofs rely heavily on results from convex analysis.


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Arseni Seregin. Jon A. Wellner. "Nonparametric estimation of multivariate convex-transformed densities." Ann. Statist. 38 (6) 3751 - 3781, December 2010.


Published: December 2010
First available in Project Euclid: 30 November 2010

zbMATH: 1204.62058
MathSciNet: MR2766867
Digital Object Identifier: 10.1214/10-AOS840

Primary: 62G07 , 62H12
Secondary: 62G05 , 62G20

Keywords: consistency , log-concave density estimation , lower bounds , maximum likelihood , mode estimation , nonparametric estimation , qualitative assumptions , shape constraints , strongly unimodal , unimodal

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 6 • December 2010
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