Wavelet regression with long memory infinite moving average errors
For more than a decade there has been great interest in wavelets and wavelet-based methods. Among the most successful applications of wavelets is nonparametric statistical estimation, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, we consider the wavelet-based estimators of the mean regression function with long memory infinite moving average errors, and investigate the 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, where the 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 demonstrate explicitly the extraordinary local adaptability of this method in handling discontinuities. We illustrate the theory with numerical examples. A technical result in our development is the establishment of Bernstein-type exponential inequalities for infinite weighted sums of i.i.d. random variables under certain cumulant or moment assumptions. These large and moderate deviation inequalities may be of independent interest.
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