14.8 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 $ L_2$-loss of $ f^*$ is not greater than the average loss of the $ f^{(s_i)}$'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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