On the Method of Internal Least Squares

Abstract

Consider a nonlinear relation y(x) = f(x, 0) + e(x) in which y(x) is a random variable observed at an independent scalar mathematical variable x; f is a known function of x; o = (01, 02, ''', 02m+q) iS a vector of unknown parameters, and e(x) is an unobserved random error variable. The problem of estimating the parameters by numerical methods has been considered by many authors, and the literature on computing algorithms has grown enormously in the past decades. Most of these algorithms need initial starting values of the unknown parameters. Good initial values play a very important role in reducing the number of iterations and obtaining least squares estimates. Poor choice of initial estimates may result in a large number of iterations before a satisfactory solution is reached, and the resulting limit estimate may not be a least squares solution. In 1948, Hartley proposed the method of internal least squares to obtain initial estimates of parameters from data. It was developed for those situations where a nonlinear regression law can be generated from an ordinary linear differential equation, and measurements on the independent variable are equally spaced. A number of authors (Hartley 1959, Patterson and Lipton 1959, Shah and Khatri 1963, and Shah 1965) have investigated the efficiency of the estimators obtained by the method of internal least squares and found that they are about 90Yo efficient compared to those obtained by the method of least squares. These findings suggest that the method of internal least squares provides very good starting values for iterative procedures. The obvious merit of the method is that it helps to obtain least squares estimates with a smaller number of iterations and, therefore, reduces the effort of finding good starting values and the computing cost. The purpose of this paper is to demonstrate that the method can be extended to unequally spaced measurements as well.