On Wavelet Regression with Long Memory Infinite Moving Average Errors

We consider the wavelet-based estimators of mean regression function with long memory infinite moving average errors and investigate their asymptotic rates of convergence of estimators based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large class of non-smooth functions that involve many jump discontinuities, whose number of discontinuities may grow polynomially fast with sample size. Therefore, in the presence of long memory moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability in handling discontinuities. A key result in our development is to establish a Bernstein-type exponential inequality for an infinite weighted sums of i.i.d. random variables under certain cumulant assumption. This large deviation inequality may be of independent interest. Short title: Wavelet Estimator with Long Memory Data 2000 Mathematics Subject Classification: Primary: 62G07; Secondary: 62C20