Symbolic cumulant calculations for frequency domain time series

With time series data, there is often the issue of finding accurate approximations for the variance of such quantities as the sample autocovariance function or spectral estimate. Smith and Field (1993) proposed a variance estimate motivated by resampling in the frequency domain. In this paper we present some results on the cumulants of this and other frequency domain estimates obtained via symbolic computation. The statistics of interest are linear combinations of products of discrete Fourier transforms. We describe an operator which calculates the joint cumulants of such statistics, and use the operator to deepen our understanding of the behaviour of the resampling based variance estimate. The operator acts as a filter for a general purpose operator described in Andrews and Stafford (1997).