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Strong \(L^p\) convergence of wavelet deconvolution density estimators. (English) Zbl 1380.42029

Summary: Using compactly supported wavelets, E. Giné and R. Nickl [Ann. Probab. 37, No. 4, 1605–1646 (2009; Zbl 1255.62103)]obtain the optimal strong \(L^\infty(\mathbb{R})\) convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Bernoulli 16, No. 4, 1137–1163 (2010; Zbl 1207.62082). This paper considers the strong \(L^p(\mathbb{R})(1 \leq p \leq \infty)\) convergence of wavelet deconvolution density estimators. We first show the strong \(L^p\) consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong \(L^p\) convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and \(p = \infty\), our convergence rate is close to Giné and Nickl’s.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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