## Year

1998

## Degree Name

Doctor of Philosophy

## Department

School of Mathematics and Applied Statistics

## Recommended Citation

Mvoi, Neofitas Sifa, The asymptotic quasi-likelihood and its application on linear models, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 1998. https://ro.uow.edu.au/theses/2062

## Abstract

This thesis is concerned with the development of the asymptotic quasi-likelihood methodology in a fixed sample space with regard to the estimation of parameters of a linear model. The asymptotic quasi-likelihood method is a statistical inference method based on the estimating functions approach. An outline of the general estimating functions theory is given.

The error term of the adopted linear model is a martingale difference. No knowledge of the distributional form of the martingale difference is assumed. The conditional variance of the martingale difference is estimated by a function derived the square of the observations. This enables the construction of the asymptotic quasi-score function from which the estimates of the parameters of the linear can be derived.

Conditions under which the root of the asymptotic quasi-score function lies in the parameter space are established. This root is shown to be a consistent estimator of the parameter of interest under these conditions. Simulation studies are carried-out and comparison is made between the least squares method and the asymptotic quasi-likelihood method.

The asymptotic quasi-score function is shown to have a limiting normal distribution under certain conditions. Asymptotic confidence intervals and hypotheses tests for the parameters are derived from this distribution. Results of analysis simulated data as well as real-life data support this convergence.

The conditional variance of the martingale difference is estimated by a function derived from the square of observations. This function is shown to be a generalization of the variance functions commonly encountered in the literature. This eliminates the problem of mis-specification of the variance function, a possible shortcoming the weighted least squares method. A graphical technique of selecting the proper estimator of the conditional variance of the martingale difference is developed. Further, a nonconstant variance diagnostic based on a chi-squared test is incorporated as a selection criterion for the proper estimator of the conditional variance martingale difference.

This procedure is used in the analysis and estimation of parameters in linear regression and experimental design. It is shown that, using this method, comparison of treatment effects of an experimental design is possible even when the error of the model has nonconstant variance.

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**