Adaptive Wavelet Estimator for Nonparametricdensity Deconvolution

Hence the problem of estimating g in (1.2) is called a deconvolution problem. The problem arises in many applications [see, e.g., Desouza (1991), Louis (1991), Zhang (1992)] 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 (1.2) using a Fourier transform [see Carroll and Hall (1988), Devroye (1989), Diggle and Hall (1993), Efromovich(1997), Fan (1991a, c), Liu and Taylor (1989), Masry (1991, 1993a, b), Stefansky (1990), Stefansky and Carroll (1990), Taylor and Zhang (1990), Zhang (1990)]. Fan (1991a, 1993) proved that the estimators of g θ are asymptotically optimal pointwise and globally, if the kernel has a limited bandwidth, that is, the Fourier transform of the kernel has bounded support. The estimators based on the deconvolution of kernel estimators and similar methods were studied in many different contexts: the asymptotic normality was established [see, e.g., Fan (1991b), Piterbarg and Penskaya (1993), Masry (1993a)]; the case of dependent εi was examined [Masry (1991, 1993b)], etc. This present paper deals with the estimation of a deconvolution density using a wavelet decomposition. The underlying idea is to present g θ via a