EXPERIENCES WITH THE BRILLINGER SPECTRAL ESTIMATOR APPLIED TO SIMULATED IRREGULARLY OBSERVED PROCESSES

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Abstract
Shannon interpolation is used to assign values from a readily simulated discrete time process to the times of a point process, simulated by Ogata's thinning technique. The result is a set of unequally spaced samples from a hypothetical continuous time process with spectrum equal to that of the discrete time process for frequencies |ω| ≤π/Δ and identically equal to zero for |ω| > π/Δ, where Δ is the discrete time step. The spectra are theoretically known both for the sampled process and for the sampling point process. We calculate Brillinger spectral estimates for examples of a process with autoregressive spectrum, sampled at the times of a Hawkes Self Exciting Point Process. The success of the Brillinger estimator is demonstrated but it is shown to have an inherently high variance. An approximate confidence interval is discussed.

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