Nonparametric estimation of multivariate convex-transformed densities

TL;DR

This paper develops nonparametric methods for estimating multivariate densities of the form p(x)=h(g(x)), where g is convex and h is a monotone function, extending log-concave density estimation to broader classes.

Contribution

It investigates the existence and consistency of maximum likelihood estimators for these convex-transformed densities, including cases with heavier tails and various monotone transformations.

Findings

MLE exists for a wide range of h functions.

MLE is consistent under general conditions.

Provides asymptotic minimax lower bounds for estimation accuracy.

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.