ABSTRACT
Suppose that given the regression model
yi =f (ti)+ 0- zi ,i= 1,2,3,...,n
where f (ti) ) is an unknown function , o is the known standard
deviation of the noise, zi are independent and identically standard Gaussian random variables. One of the problem in regression analysis is how to estimate the regression function f ( ti ) with small risk
n
R( ,f ) = âEE(fi âfi)2.
n i =1
Donoho and Johnstone (1994) have developed a powerful methodology based on the principle of shrinking wavelet coefficients towards zero. Their procedure is called WaveShrink.
The computtationally efficient formula for computing the exact risk of WaveShrink estimates in finite sample situations can be derived. From this formula, the behaviour of WaveShrink estimator such as the accuraty of two different fuction models such as Doppler and Heavisine can be observed.
Using S+Wavelets computer simulation for our different fuctions models can be observed that
(1) significantly WaveShrink using soft shrinkage and the universal threshold have the biggest risk, and it is not appropriate to apply the universal threshold when using soft shrinkage in WaveShrink.
(2) WaveShrink using hard shrinkage have smaller risk than WaveShrink using soft shrinkage for all sample size.
(3) for all sample size, the risk of the Waveshrink estimate decrease as sample size increase
Keywords: nonparametric regression, wvelet transform, WaveSrin
