Standard wavelet-based density estimators may not retain some global properties of the curve, e.g. non-negativity and integral equal to one. This has the additional disadvantage in small samples of mass being taken out from "right places" and being put on "inappropriate places", e.g. below the X-axis. We present a class of new wavelet-based estimation methods intended to retain asymptotic minimax optimality rates of standard methods by achieving non-negativity in a natural way. Moreover, the choice of the threshold level in the estimation process can be made in a simple adaptive manner. Basic for our procedure is the presentation of the density f ( x ) as a trace of an appropriate multivariate function expanded in a wavelet series. First, a new nonlinear approximation off (x) is proposed. The empirical version of the approximation yields the estimator. The estimators of the wavelet coefficients are also of non-linear type.
This article is a systematic overview of compression, smoothing and denoising techniques based on shrinkage of wavelet coefficients, and proposes (in Sections 5 and 6) an advanced technique for generating enhanced composite wavelet shrinkage strategies.
\Ve introduce a general class of shape-preserving wavelet approsimating operators (approximators) which transform cumulative distribution functions and densities into functions of the same type. Our operators can be considered as a generalization of the operators introduced by Anastassiou and Yu [I]. Further, we extend the consideration by studying the approximation properties for the whole variety of L,-norms, 0 < p 5 m. In [l] the case p = o is discussed. Using the properties of integral moduli of smoothness, we obtain various approximation rates under no (or minimal) additional assumptions on the functions to be approximated. These assumptions are in terms of the function or its Riesz potential belonging to certain ho~nogeneous Besov. Triebel-Lizorkin. Sobolev spaces, the pace BT., of functions ~v i t h bounded Wiener-Young p-variation, etc.
PRELIMISARIES
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