On the estimation of common non-linearity among repeated time series

The bispectrum. is a higher-order statistic and is known to be a useful tool for detecting non-linearity. A succinct example of its power to identify non-linear sound waves from broken bridge struts was given by Rivola and White (1998). As well as detecting non-linearity it has the further advantage that its magnitude and shape can be used to estimate the third order non-linear structure (Barnett and Wolff). When a time series is repeated (such as sound waves from a collection of bridge struts) Diggle and Al-Wasel (1997) showed how to produce a common spectrum and to estimate individual departures from this global quantity. The purpose of this paper is to extend this method to the bispectrum and give a summary of common non-linearity among repeated time series. We evaluate our method using data from a group of people speaking the letter 'A' and from one person repeatedly speaking this letter.