MULTIRESOLUTION ANALYSES AND WAVELETS FOR DENSITY ESTIMATION

Abstract. We use the concept of multiresolution analysis and orthonormal basisof wavelet functions to construct an estimator for an unknown density in a givenSobolev’s space. We show consistency with the rate of convergence equal to theoptimal minimax rate of convergence. We also discuss some of the features of thisnew method of estimation on a numerical example. 1. IntroductionLet X 1 ,X 2 ,··· ,X n be independent identically distributed random variables, thecommon distribution having the density f with respect to the Lebesque measure ofIR d . Many methods for the estimation of the unknown density f have been proposedand used over the years. They include the kernel methods (Rozenblatt [14], Parzen[13]. Watson and Leabetter [22], the orthogonal series estimators used by Cencov [3],Kronmal and Tarter [10], Schwartz [16], Walter ([19],[20]), Walter and Blum [21], theinterpolation methods considered by Wahba ([17], [18]). We refer the reader to theextensive bibliographies of the textbooks [5] and [15] on density estimation.The kernel method of density estimation is presumably the most popular of all ofthem. Its local character is certainly its main attraction. The method of orthogonal

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