Automatic Smoothing of the Log Periodogram

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Abstract
We develop an objective optimum smoothing procedure for an estimate of the log spectral density, based on smoothing the log periodogram with a smoothing spline. We call the result the optimally smoothed spline (OSS) estimate. We show that an unbiased estimate [Rcirc](λ) of the expected integrated mean square error can be obtained as a function of the smoothing, or “bandwidth” parameter λ. The smoothing parameter then is chosen as that λ that minimizes [Rcirc](λ). The degree of the smoothing spline (equivalently, the “shape” parameter of the window) also can be chosen this way. Results of some Monte Carlo experiments demonstrating the effectiveness of the method on selected examples are given.

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