Coupled and separable iterations in nonlinear estimation

Abstract

This thesis deals with algorithms to fit certain statistical models. We are concerned with the interplay between the numerical properties of the algorithm and the statistical properties of the model fitted. Chapter 1 outlines some results, concerning the construction of tests and the convergence of algorithms, based on quadratic approximations to the likelihood surface. These include the relationship between statistical curvature and the convergence of the scoring algorithm, separable regression, and a Gauss-Seidel process which we called coupled iterations. Chapters 2, 3 and 4 are concerned with varying parameter models. Chapter 2 proposes an extension of generalized linear models by including a linear predictor for (a function of) the dispersion parameter also. Chapter 3 deals with various ways to go outside this extended generalized linear model framework for normally distributed data. Chapter 4 briefly describes how coupled iterations may be applied to autoregressive and multinormal models. Chapters 5 to 8 apply a generalization of Prony's classical parametrization to solve separable regression problems which satisfy a linear homogeneous difference equation. Chapter 5 introduces the problem, specifies the assumptions under which asymptotic results are proved, and shows that the reduced normal equations may be expressed as a nonlinear eigenproblem in terms of the Prony parameters. Chapter 6 describes the algorithm which results from solving the eigenproblem, including some computational details. Chapter 7 proves that the algorithm is asymptotically stable. Chapter 8 compares the convergence of the algorithm with that of Gauss-Newton by way of simulations.