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∈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∈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.