Nonparametric Estimation of Multivariate Convex-Transformed Densities

Tech Report Number
562

 

Abstract

We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ R d and for a fixed 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 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 classes of densities with heavier tails than the log-concave class. We first investigate when the MLE pbexists 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 MLE 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 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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