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 [Formula: see text]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 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 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.