The estimation of the bispectral density function and the detection of periodicities in a signal

In a recent paper Subba Rao and Gabr (J. Time Ser. Anal. (1987), in press) considered the estimation of the spectrum and the inverse spectrum based on the method by Pisarenko (Geophys. J. Roy. Astronom. Soc. 28 (1972), 511-531). The asymptotic properties of these estimates were studied using the properties of Wishart matrices. In this paper we show how the method can be extended to the estimation of the bispectral density function, an important tool in the study of non-Gaussian time series. All these methods of estimation are illustrated with simulated examples. In the illustrations considered, the emphasis is on the detection of periodicities in the "signal" (possibly in the presence of noise). We also considered an example based on real data. These data arise in the study of the earth's magnetic reversals and the detection of periodicities.