Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series
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.
scientific article; zbMATH DE number 1134827
scientific article