Standard wavelet shrinkage procedures for nonparametric regression
are restricted to equispaced samples. There, data are transformed into
empirical wavelet coefficients and threshold rules are applied to the
coefficients. The estimators are obtained via the inverse transform of the
denoised wavelet coefficients. In many applications, however, the samples are
nonequispaced. It can be shown that these procedures would produce suboptimal
estimators if they were applied directly to nonequispaced samples.
¶ We propose a wavelet shrinkage procedure for nonequispaced samples.
We show that the estimate is adaptive and near optimal. For global estimation,
the estimate is within a logarithmic factor of the minimax risk over a wide
range of piecewise Hölder classes, indeed with a number of
discontinuities that grows polynomially fast with the sample size. For
estimating a target function at a point, the estimate is optimally adaptive to
unknown degree of smoothness within a constant. In addition, the estimate
enjoys a smoothness property: if the target function is the zero function, then
with probability tending to 1 the estimate is also the zero function.