The mean consistency of wavelet density estimators

The wavelet estimations have made great progress when an unknown density function belongs to a certain Besov space. However, in many practical applications, one does not know whether the density function is smooth or not. It makes sense to consider the mean L p $L_{p}$ -consistency of the wavelet estimators for f ∈ L p $f\in L_{p}$ ( 1 ≤ p ≤ ∞ $1\leq p\leq\infty$ ). In this paper, the authors will construct wavelet estimators and analyze their L p ( R ) $L_{p}(\mathbb{R})$ performance. They prove that, under mild conditions on the family of wavelets, the estimators are shown to be L p $L_{p} $ ( 1 ≤ p ≤ ∞ $1\leq p\leq\infty$ )-consistent for both noiseless and additive noise models.