A Monte Carlo Study of Non-Linear Regression Theory

Multiple regression provides the capability of using non-linear functions to fit various curvilinear surfaces. These non-linear functions are, however, linear in the parameters. Non-linear term of the variables such as x2 , x2 , ln X, X, YX are incorporated in a linear model. For example: Y = b0 + b1x1 +b2x2 + b3lnx2 + ϵ Many practical situations require the fitting of mathematical functions which are non-linear in the parameters and perhaps the variables. For example: Y = b, eb2x + ϵ The Modified Gauss-Newton Method (Hartley, 1961) provides a solution to fitting models of this kind to data situations where the deviations from the model may be appreciable. The solution requires an iterative procedure which is straight forward from a theoretical point of view, but for which many problems may arise computationally. The laborious computations are best carried out using a high speed computer. The several computor programs that have been developed (Gauss, 1957; Booth and Peterson, 1958; and Marquardt, 1964) all have the failing of blowing up with certain data sets and models. This generally takes the form of an exponent underflow or overflow (see Note 6). Excessive numbers of iterations may also be required. The statistical properties of the estimators have been undertain. Hartley and Booker (1965) give equations for the variances and covar­iances of the estimators which are said to be asymptotically correct. This paper concerns the efficiency of Hartley\u27s modifications. Some extra modifications which have proved necessary are also derived. The Monte Carlo study of the distributional properties of the estimators is discussed