Maximum Likelihood Estimation by Means Op Nonlinear Least Squares

Abstract

Methods are given for using readily available nonlinear regression programs to produce maximum likelihood estimates in a rather natural way. Used as suggested the common Gauss‐Newton algorithm for nonlinear least squares becomes the Fisher scoring algorithm for maximum likelihood estimation. In some cases it is also the Newton‐Raphson algorithm. The standard errors produced are the information theory standard errors up to a possible common multiple. This means that much of the auxiliary output produced by a nonlinear least squares analysis is directly applicable to a maximum likelihood analysis. Illustrative applications to Poisson, quantal response, multinomial, and log‐linear models are given.

Multidimensional successive categories scaling: A maximum likelihood method

“…Model (2) postulates that the error is additive, and is normally distributed over 2 (within each subject). This error replications of judgments with constant variance ok model was implicit in the initial scaling phase of classical MDS [Torgerson, 1952].…”

Estimation and Interpretation of a Nonlinear Migration Model

“…Let the expectation of y be p and the covariance matrix of y be A. Two remarkable properties of the regular exponential likelihood function have been derived by Jennrich and Moore (1975 …”

“…Note that where the number of degrees of freedom V equals the number of elements in y minus the number of elements in 6. The motivations for this correction are that the results are analogous to the standard errors in nonlinear least squares problems, and that it does not affect the nice asymptotic properties, because S2 approaches one as the sample size approaches infinity (Jennrich and Moore 1975). Note that without this correction, a variable that contributes practically nothing to the reduction in S2 is sometimes found to have a t-ratio of large magnitude, say, about 4 or 5.…”

Multiple-Spell Regression Models for Duration Data

“…The program utilizes the Gauss-Newton method for non-linear least squares estimation by modifying the scoring algorithm to accommodate maximum likelihood problems; see Jennrich and Moore (1975) or Petersen (1986b) for details. The parameter estimates and their estimated standard deviations are given in Table 1. The calculation of the 'residuals' ri = -log [S(ti; zi; 0, ,B)] (see, for example, Cox andOakes (1984), p. 89, or Kalbfleisch andPrentice (1980), p. 96) of model 1 and plotting r against the logarithm of the proportion of residuals exceeding r shows a good fit to a straight line with slope -1.…”