A GENERALIZATION OF THE GAUSS-NEWTON METHOD THAT SOLVES EXTENDED LEAST SQUARES PROBLEMS1

Abstract. Modeling the mean of a random variable as a function of unknown parameters leads to a nonlinear least-squares objective function. The Gauss-Newton method reduces nonlinear least-squares problems to a sequence of linear least-squares problems and requires only first-order information about the model functions. In a more general heteroscedastic setting, there are also unknown parameters in a model for the variance. This leads to an objective function that is no longer a sum of squares. We present an extension of the Gauss-Newton method that minimizes this objective function by reducing the problem to a sequence of linear least-squares problems and requires only first-order information. This represents a new result because other methods that reduce this problem to a sequence of linear least-squares problems do not necessarily converge. Key words. Iteratively reweighted least squares, extended least squares, asymp-totic statistic