Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series
- Neumann, Michael H.
2010 Mathematics Subject Classification
- 62M15 62M10 62G07
- Spectral density estimation, wavelet estimators, asymptotic normality, large deviations
In the present paper we consider nonlinear wavelet estimators of the spectral density ƒ of a zero mean stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of estimators of coefficients that arise from a Fourier series expansion according to a wavelet basis. The main goal of this paper is to prepare the ground for the application of these methods to spectral density estimation, which is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. For that we derive upper estimates for their cumulants, which yield the asymptotic normality in terms of probabilities of large deviations. Using these results we can conclude the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. Hence, we obtain estimators of ƒ, which keep all interesting properties like high spatial adaptivity that are already known from wavelet estimators in the case of Gaussian regression. It turns out that optimally tuned versions of these estimators attain the optimal uniform rate of convergence of their L2-risk in a wide variety of Besov smoothness classes.