Convergence Rates of Multivariate Density Estimation Methods Based on Adaptive Partitioning
Authors: Linxi Liu, Dangna Li, Wing Hung Wong; 24(50):1−64, 2023.
Abstract
Density estimation is a fundamental component of various statistical methods, including classification, nonparametric testing, and data compression. This paper focuses on the non-parametric approach to multivariate density estimation and investigates its asymptotic properties in both frequentist and Bayesian settings. The density function is estimated by considering a sequence of approximating spaces that consist of piecewise constant density functions supported by binary partitions of increasing complexity. The estimation is performed by maximizing the likelihood of the corresponding histogram on the partition or the marginal posterior probability of the partition under a suitable prior. The convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator are analyzed. It is concluded that for a relatively rich class of density functions, the convergence rate does not directly depend on the dimension. Additionally, the Bayesian method is shown to adapt to the unknown smoothness of the density function. The method is applied to specific function classes, including spatially sparse functions, functions of bounded variation, and H{\\\”o}lder continuous functions, and explicit rates are obtained. An ensemble approach is introduced, which aggregates multiple density estimates fitted under carefully designed perturbations. It is demonstrated that for density functions belonging to a H{\\\”o}lder space ($\\mathcal{H}^{1, \\beta}, 0 < \\beta \\leq 1$), the ensemble method can achieve a minimax convergence rate up to a logarithmic term, while the convergence rate of the density estimator based on a single partition is suboptimal for this function class.
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