Abstract
In this paper, multiwavelet deconvolution density estimators are presented by a linear multiwavelet expansion and a nonlinear multiwavelet expansion, respectively. Moreover, the unbiased estimation is shown, and asymptotic normality is discussed for the multiwavelet deconvolution density estimators. Finally, a numerical example is given for our discussion.
Highlights
Introduction and PreliminaryAssume that (Ω, F, P) is a probability space
Yn) can be recognized as wavelet estimator fn means a deconvolution that fn can be expanded by a wavelet basis
Note: A ≲ B denotes two variables A, B satisfying A ≤ cB, for some constant c > 0; A≳B is equivalent to B ≲ A, and A ∼ B means both A ≲ B and A≳B
Summary
Assume that (Ω, F, P) is a probability space. Yn) can be recognized as wavelet estimator fn means a deconvolution that fn can be expanded by a wavelet basis. Choose a multiscaling function Φ with multiplicity r satisfying the following condition:. Note: A ≲ B denotes two variables A, B satisfying A ≤ cB, for some constant c > 0; A≳B is equivalent to B ≲ A, and A ∼ B means both A ≲ B and A≳B. The density function fε of the random noise ε satisfies the following conditions [2]:. (C2) |fFε T(ω)|≳(1 + |ω|2)− (β/2) (C3) |(fFε T)(m)(ω)|(1 + |ω|2)− (β+m/2), m 0, 1, 2 Under these two conditions, the random noise ε is said to be ill-posed
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