Li, Z; (2014) Methods for Irregularly Sampled Continuous Time Processes. Doctoral thesis , UCL (University College London).

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Abstract

This thesis will consider methods associated with irregularly spaced sampling of a real-valued continuous time stationary process. The problem of Monte Carlo simulation as well as parametric estimation under irregularly spaced sampling times will be discussed. For the simulation problem, the focus will be on the spectral simulation method. A novel algorithm has been proposed for the determination of the spectral simulation scheme, which is optimal in the sense of achieving required accuracy with minimal computational costs. The problem of parametric estimation under irregularly spaced sampling times will also be discussed. We will adapt the framework stochastic sampling times, in which the irregularity of the sampling times is modeled through a renewal point process over the real line. By constructing a second order discrete time stationary process from sampling, a parametric estimation method based on the well-known Whittle log-likelihood function will be proposed. Asymptotic consistency of the resulting estimator will be proved by borrowing existing results from literature of renewal theory. Moreover the performance issue of this proposed estimation procedure will be investigated further. It will be shown that by calculating the spectral density of the sampled discrete time process through a Discrete Fourier Transform (DFT) approximation, the Whittle log-likelihood function can indeed be evaluated relatively efficiently. This estimation method, however, will induce information loss, which will be shown to be related to the unique properties of the renewal kernel function. Although a accurate analysis of the renewal kernel function is not easy, it is still possible to provide some insights on the determining factors of the information loss through asymptotic calculations.

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