Automatic estimation of the cross-spectrum of a bivariate time series

SUMMARY The penalised Whittle likelihood has recently been shown to have good properties in nonparametric estimation of spectral density functions. This paper extends the approach to the estimation of the cross-spectrum of a bivariate time series. One major difference from the univariate case is that the cross-spectrum estimate is not constrained to be positive, but must result in a positive definite spectral density matrix. An efficient computational method based on iterative reweighted least-squares is described, and an estimate of the integrated squared-error loss is derived and used as an objective criterion to allow automatic selection of the smoothing parameters. Numerical experiments indicate that the proposed estimate improves on the standard estimate based on kernel smoothing of the cross-periodograms. An analysis of respiration and heart rate time series is given as an illustrative example.