Convergence Rates for Estimating Multivariate Scale Mixtures of Uniform Densities

• Activity Number: 321 - Nonparametric Inference Under Shape Constraints
• Type: Topic Contributed
• Date/Time: Tuesday, August 9, 2022 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Adityanand Guntuboyina*
• Companies: University of California Berkeley
• Keywords: Density estimation; Bracketing entropy; Curse of dimensionality; Piecewise constant estimator

Abstract

The Grenander estimator is a very important and well-studied procedure for univariate nonparametric density estimation. It is usually defined as the Maximum Likelihood Estimator over the class of all non increasing densities on the positive real line. It can also be seen as the Maximum Likelihood Estimator over the class of all scale mixtures of uniform densities. Using the latter viewpoint, Pavlides and Wellner proposed a multivariate extension of the Grenander estimator as the nonparametric maximum likelihood estimator 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. This is joint work with Arlene K. H. Kim (Korea University) and Frank Fuchang Gao (University of Idaho).