Posterior probability intervals for wavelet thresholding
Stuart Barber, Guy P. Nason & Bernard W. Silverman.
We use cumulants to derive Bayesian credible intervals for wavelet
regression estimates. The first four cumulants of the posterior
distribution of the estimates are expressed in terms of the observed
data and integer powers of the mother wavelet functions. These powers
are closely approximated by linear combinations of wavelet scaling
functions at an appropriate finer scale. Hence, a suitable
modification of the discrete wavelet transform allows the posterior
cumulants to be found efficiently for any given data set. Johnson
transformations then yield the credible intervals themselves.
Simulations show that these intervals have good coverage rates, even
when the underlying function is inhomogeneous, where standard methods
fail. In the case where the curve is smooth, the performance of our
intervals remains competitive with established nonparametric
Some key words:
Bayes estimation; Cumulants; Curve
estimation; Interval estimates; Johnson curves; Nonparametric
regression; Powers of wavelets.
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