Commonly used wavelet estimators are, in contrast to kernel
estimators, not translation-invariant: if we shift the underlying data
set by a small amount , apply nonlinear thresholding and shift the
estimator back by
, then this new estimator
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
, the
following new estimator
.
This estimator possesses some advantages over the usual
estimation scheme. First, it follows immediately by Jensen's
inequality that the -loss of
is not greater than the average
loss of the
'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.
In our example the number of shifts is always with
the
number of observations.
The interactive menu provides you the opportunity to further improve the estimate.