Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series
type: Item
instance of: scholarly article
title: Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series
zbMATH Open document ID: 0885.62046
DOI: 10.1016/S0304-4149(96)00005-1
published in: Stochastic Processes and their Applications
publication date: 1998-03-29T00:00:00Z
review text: The present paper deals with the nonparametric estimation of a multivariate density function using the wavelet method. The representation of a smooth function through wavelets is recalled and the estimate of a \(d\)-variate (\(d\geq 1\)) density function is presented. The main result consists in formulation and proof of a theorem establishing the strong consistency of the estimate and showing that the optimal rate of convergence is attained, similarly to other techniques of density/regression function estimation; namely, the best rates given by \textit{D.L. Donoho} et al. [Ann. Stat. 24, No. 2, 508-539 (1996; Zbl 0860.62032)], and in the case of density with Lipschitz-continuous derivatives the rates given in classical papers, e.g. of \textit{C.J. Stone} [Ann. Stat. 10, 1040-1053 (1982; Zbl 0511.62048)]). The author considers a joint density of a part (of length \(d\)) of a stationary random process, such that the corresponding stationary \(d\)-dimensional process is a strongly mixing one. The paper is highly technical. Main properties of wavelets and of multiresolution analysis are briefly presented. The second half of the paper contains the proofs of the main results and of several connected lemmas.
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62H12
zbMATH DE Number: 1134827
zbMATH Keywords: density estimation
zbMATH Keywords: wavelet method
zbMATH Keywords: rate of convergence
zbMATH Keywords: strong consistency
author: Elias Masry
reviewed by: Petr Volf
MaRDI profile type: Publication
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full work available at URL: s0304-4149(96)00005-1
OpenAlex ID: W2086560190
description: scientific article; zbMATH DE number 1134827