##
**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.

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 | Asymptotic properties of nonparametric inference |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

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\textit{M. Pensky} and \textit{B. Vidakovic}, Ann. Stat. 27, No. 6, 2033--2053 (1999; Zbl 0962.62030)

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