Nonparametric estimation of multivariate convex-transformed densities

Abstract

We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$ for $x\in {\mathbb{R}}^d$ and for a fixed monotone function $h$ and an unknown convex function $g$. The canonical example is $h(y)=e^{-y}$ for $y\in \mathbb{R}$; in this case, the resulting class of densities [\mathcal {P}(e^{-y})={p=\exp(-g):g 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 $\hat{p}$ 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 $x_0$ under natural smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from convex analysis.

Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Inference and Modeling with Log-concave Distributions
• Nonparametric estimation of multivariate convex-transformed densities
• Quasi-concave density estimation

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• Optimal rates of convergence for convex set estimation from support functions
• A Bayesian nonparametric approach to log-concave density estimation
• Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its Consistency