The exponential regression curve and other non-linear approximations.

This thesis is concerned with the estimation of the nonlinear parameters in statistical models consisting of a linear combination of exponential terms and an error term from series of observations taken at equi-spaced intervals. For such models, with normally and independently distributed errors the maximum likelihood and the least squares equations are complicated and soluble only by iteration. These complications increase with the number of the non-linear parameters. Some previous work has been done by Patterson and Taylor on estimating the non-linear parameter for the one-exponential regression equation and by Cornell on estimating the non-linear parameters for k exponentials, by means of a direct approach. The present contribution is a continuation of this work. In chapter 1 we discuss the possibility of reducing the number of the maximum likelihood equations to a set of equations involving the non-linear parameters only, and then solving these equations by the Newton-Raphson method. We also suggest an alternative procedure for solving the maximum likelihood equation of the one-exponential-term model. Two numerical examples are introduced for illustration. In chapter 2 we propose a new general method for estimating the non-linear parameters of k exponentials. It is shown that the estimators are consistent. Asymptotic formulae for the bias and the variance for the estimator of the non-linear parameter and confidence limits for the parameter are deduced in the case of the two particular models yi = betapi + ei , yi= alpha + betapi + ei . The estimator used is a ratio of linear combinations of the observations. In order to examine its exact sampling distribution, we deduce the formula for the probability density function of the ratio of two normal variates. The form of the frequency curve is Illustrated by a series of numerical examples. Chapter 3 shows the applications of the method to a number of cases. The closeness of fit to the observations and the accuracy of the estimates are compared with the estimates given by Cornell's general methods. Also a brief comparison is made with Patterson's and Taylor's estimators of the non-linear parameter of the one-exponential regression curve. In chapter 4 we discuss the method of minimum deviations and its applications to the non-linear models