8. Translation Invariance

Commonly used wavelet estimators are, in contrast to kernel estimators, not translation-invariant: if we shift the underlying data set by a small amount s, apply nonlinear thresholding and shift the estimator back by s, then this new estimator f(s) is usually different from the estimator without the shifting and backshifting operation. Coifman and Donoho (1995) and Nason and Silverman (1994) proposed to make the wavelet estimators translation-invariant and defined, with shifts $s_i,\ i=1,\dots,I$, the following new estimator $f^*(x)=\sum_i f^{(s_i)}(x)/I$.

This estimator possesses some advantages over the usual estimation scheme. First, it follows immediately by Jensen's inequality that the L2-loss of f* is not greater than the average loss of the f(si)'s. Second, wavelet estimators sometimes have a quite irregular visual appearance. Often there are some spurious features caused by random fluctuations. This effect is weakened by averaging over different shifts as described above. In a small simulation, Neumann (1996) observed a considerable improvement over the standard estimation scheme, even by taking only a small number of shifts.


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In our example the number of shifts is always $\log_2(n)$ with n the number of observations.

The interactive menu provides you the opportunity to further improve the estimate.


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