Nonparametric estimation of multivariate convex-transformed densities

The Annals of Statistics · 2010 · Volume 38 · Issue 6

Abstract

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)=e−y 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 p̂ 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.

Keywords

62G05, 62G07, 62G20, 62H12, consistency, log-concave density estimation, lower bounds, maximum likelihood, mode estimation, nonparametric estimation, qualitative assumptions, shape constraints, strongly unimodal, unimodal

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