A simple generalised crossvalidation method of span selection for periodogram smoothing

A consistent estimator for the spectral density of a stationary random process can be obtained by smoothing the periodograms across frequency. An important component of smoothing is the choice of the span. Lee (1997) proposed a span selector that was erroneously claimed to be unbiased for the mean squared error. The naive use of mean squared error has some important drawbacks in this context because the variance of the periodogram depends on its mean, i.e. the spectrum. We propose a new span selector based on the generalised crossvalidation function derived from the gamma deviance. This criterion, originally developed for use in fitting generalised additive models, utilises the approximate full likelihood of periodograms, which asymptotically behave like independently distributed chi-squared, i.e. gamma, random variables. The proposed span selector is very simple and easily implemented. Simulation results suggest that the proposed span selector generally outperforms those obtained under a mean squared error criterion.