Convergence rates for estimating multivariate scale mixtures of uniform densities

TL;DR

This paper establishes that a multivariate extension of the Grenander estimator for scale mixtures of uniform densities converges at a near-univariate rate, partially overcoming the curse of dimensionality, and confirms a conjecture under certain conditions.

Contribution

It proves the convergence rate of the multivariate Grenander estimator and confirms a conjecture, providing theoretical insights and practical algorithms for estimation.

Findings

Achieves near-univariate cube root convergence rate in multivariate setting

Avoids full curse of dimensionality for this estimator

Provides algorithms and empirical validation

Abstract

The Grenander estimator is a well-studied procedure for univariate nonparametric density estimation. It is usually defined as the Maximum Likelihood Estimator (MLE) over the class of all non-increasing densities on the positive real line. It can also be seen as the MLE over the class of all scale mixtures of uniform densities. Using the latter viewpoint, Pavlides and Wellner~\cite{pavlides2012nonparametric} proposed a multivariate extension of the Grenander estimator as the nonparametric MLE over the class of all multivariate scale mixtures of uniform densities. We prove that this multivariate estimator achieves the univariate cube root rate of convergence with only a logarithmic multiplicative factor that depends on the dimension. The usual curse of dimensionality is therefore avoided to some extent for this multivariate estimator. This result positively resolves a conjecture of Pavlides and Wellner~\cite{pavlides2012nonparametric} under an additional lower bound assumption. Our proof proceeds via a general accuracy result for the Hellinger accuracy of MLEs over convex classes of densities. We also provide algorithms for computing the estimator, and illustrate performance on real and simulated datasets.