Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood

Abstract

The penalized likelihood approach is not well developed in time series analysis, even though it has been applied successfully in a number of nonparametric function estimation problems. Chow and Grenander proposed a penalized likelihood-type approach to the nonparametric estimation of the spectral density of Gaussian processes. In this article this estimator is extended to more general stationary processes, its practical implementation is developed in some detail, and some asymptotic rates of convergence are established. Its performance is also compared to more widely used alternatives in the field. A computational algorithm involving an iterative least squares, initialized by the log-periodogram, is first developed. Then, motivated by an asymptotic linearization, an estimator of the integrated squared error between the estimated and true log-spectral densities is proposed. From this, a data-dependent procedure for selection of the amount of smoothing is constructed. The methodology is illustrated with some real and simulated data sets. A simulation study with autoregressive and moving average processes is conducted to provide quantification of the performance characteristics of the approach relative to some well-established methods in the literature, including the smoothed log-periodogram, the logarithm of the smoothed periodogram, and ARMA spectral density estimators. Empirical rates of convergence of the estimator are evaluated and shown to be well predicted by an asymptotic analysis.

Key Words:

• Adaptive smoothing
• AIC
• Cross-validation
• Iterative least squares
• Stationary time series
• Regularization