A function space approach to the generalized nonlinear model with applications to frequency domain spectral estimation
Peter McCullagh (1983) outlined the theory of quasi-likelihood estimation in generalized linear models. Chiu (1988) showed that an iterated, reweighted least squares procedure applied to the periodogram produces estimates of spectral density model parameters for Gaussian univariate time series which have the same asymptotic variance as those produced by maximizing the true likelihood. In this dissertation, McCullagh's theory is combined with a functional analysis approach and extended to parametric estimation of the spectral density matrix components of a non-Gaussian bivariate time series. An asymptotic optimality theorem is given, which shows optimality of an iterated, reweighted least squares procedure within a class of procedures. However, the principal application of the theory is for parametric spectral estimation in the case of an observed "contaminated" Gaussian series X(t)+N(t), where the noise series is uncorrelated with the X series, and it is desired to estimate the spectrum of the X series. Previous literature suggests removing contaminated bands of the periodogram prior to analysis, but the results of the dissertation may be used to unbiasedly estimate the spectrum of f without a precise knowledge of which bands are contaminated.