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Adaptive wavelet estimator for nonparametric density deconvolution. (English) Zbl 0962.62030
From the introduction: Let $\theta$ and $\varepsilon$ be independent random variables with density functions $g$ and $q$, respectively, where $g$ is unknown and $q$ is known. One observes a sample of random variables, $$X_i=\theta_i +\varepsilon_i, \quad i=1,2, \dots,n.\tag 1$$ The objective is to estimate the density function $g$. In this situation the density function $p$ of $X_i$, $i=1,\dots,n$, is the convolution of $q$ and $g$, $$p(x)= \int^\infty_{-\infty} q(x-\theta) g(\theta)d \theta. \tag 2$$ Hence the problem of estimating $g$ in (2) is called a deconvolution problem. The problem arises in many applications and, therefore, it was studied extensively in the last decade. The most popular approach to the problem was to estimate $p(x)$ by a kernel estimator and then solve equation (2) using a Fourier transform. This paper deals with the estimation of a deconvolution density using a wavelet decomposition. The underlying idea is to present $g(\theta)$ via a wavelet expansion and then to estimate the coefficients using a deconvolution algorithm. The proposed approach is based on orthogonal series methods for the estimation of a prior density, and also on modern developments of wavelet techniques in curve estimation. The estimators proposed in this paper are based on Meyer-type wavelets rather than on wavelets with bounded support. Meyer-type wavelets form a subset of the set of band-limited wavelets that allow immediate deconvolution. It should be noted that the nonlinear wavelet estimator constructed in this paper is based on a “global thresholding” which is somewhat different from the “block thresholding”: in the “global thresholding” procedure all coefficients of the same level are thresholded simultaneously, while “block thresholding” groups together only a finite number of coefficients. This article is organized in the following way. In Section 2 we give a brief description of Meyer-type wavelets and derive the linear and nonlinear wavelet estimators of $g(\theta)$. In Section 3 we investigate asymptotic behavior of the estimators when $g(\theta)\in H^\alpha$. The case of supersmooth $g(\theta)$ is considered in Section 4. In Section 5 we illustrate the theory by examples. Section 6 concludes the paper with discussion. Section 7 contains proofs of the theorems.

##### MSC:
 62G07 Density estimation 62G05 Nonparametric estimation 62G20 Nonparametric asymptotic efficiency 42C40 Wavelets and other special systems
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