Abstract
By developing the classical kernel method, Delaigle and Meister provide a nice estimation for a density function with some Fourier-oscillating noises over a Sobolev ball and over risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball and risk with by using wavelet methods. We firstly show a linear wavelet estimation for densities in over risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when . Because the linear wavelet estimator is not adaptive, a nonlinear wavelet estimator is then provided. It turns out that the convergence rate is better than the linear one for . In addition, our conclusions contain estimations for density derivatives as well.
Highlights
Introduction and preliminaryOne of the fundamental deconvolution problems is to estimate a density function fX of a random variable X, when the available data W, W, . . . , Wn are independent and identically distributed (i.i.d.) withWj = Xj + δj (j =, . . . , n).We assume that all Xj and δj are independent and the density function fδ of the noise δ is known.Let the Fourier transform f ft of f ∈ L(R) be defined by f ft(t) = R f (x)eitx dx in this paper
We find from Theorem . and Theorem . that the nonlinear wavelet estimator converges faster than the linear one for r ≤ p
This paper studies wavelet estimations of a density and its derivatives with Fourier-oscillating noises
Summary
The fundamental method to construct a wavelet basis comes from the concept of multiresolution analysis (MRA [ ]). It is defined as a sequence of closed subspaces {Vj} of the square integrable function space L (R) satisfying the following properties:. A family of important examples are Daubechies wavelets D N (x), which are compactly supported in time domain [ ]. They can be smooth enough with increasing supports as N gets large, D N do not have analytic formulas except for N =. Let h be a Daubechies scaling function or the corresponding wavelet.
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