Wavelet Shrinkage For Nonequispaced Samples
Tony Cai and Lawrence Brown
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.
Cai, T. & Brown, L.D. (1999).
Wavelet estimation for samples with random uniform design.
Statistics and Probability Letters 42, 313-321.