Convergence rates for estimating multivariate scale mixtures of uniform densities

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.

💡 Research Summary

The paper investigates the statistical properties of a multivariate extension of the Grenander estimator, which is defined as the non‑parametric maximum likelihood estimator (MLE) over the class of multivariate scale mixtures of uniform (SMU) densities. In the univariate case the Grenander estimator coincides with the MLE over the class of non‑increasing densities on (0,∞) and enjoys the optimal Hellinger‑distance rate n⁻²⁄³. Pavlides and Wellner (2012) proposed a multivariate analogue by considering the class

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