Abstract
We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.
Highlights
A nonparametric regression NPR problem consists of estimating an unknown mean function that smoothly changes between observations at different design points
Brown and Low 1 showed that the NPR experiment is asymptotically equivalent to the white-noise model where the mean function is observed in the presence of a Brownian motion process
The original asymptotic equivalence results for NPR experiments were extended by Brown et al 4 and Carter 5, 6 along with refinements in the approximations from Rohde 7 and Reiss 8
Summary
A nonparametric regression NPR problem consists of estimating an unknown mean function that smoothly changes between observations at different design points. Brown and Low 1 showed that the NPR experiment is asymptotically equivalent to the white-noise model where the mean function is observed in the presence of a Brownian motion process. Lemma 3.1 shows that a discrete wavelet transform of the observations from F produces observations with nearly the same distribution as these asymptotically sufficient statistics In both experiments the lower-frequency terms in the wavelet decomposition are sufficient for estimating the means, allowing the higher-frequency terms to be used to give information about the variance process. This result extends the sort of approximation by Donoho and Johnstone 12 to correlated errors and is very much in the spirit of our Theorem 1.1 here. The proof for Theorem 1.2 is in Section 4 with some relevant bounds in Sections 5 and 6
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