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A Family of Functional Iterations and the Solution of Maximum Likelihood Estimating Equations

Leon L. Wegge

Abstract

In this paper a family of functional iterations is introduced. One member of this family is the Newton-Raphson method and another member, obtained from a generalization of Steffensen's method to a system of equations, has been considered in [7]. The general member of the family is derived from a regula falsi construction, due to Gauss, for a particular choice of points in the iteration. From the computational point of view, all the members of the family of iterations, except the Newton-Raphson method, have the property that the partial derivatives of the system of equations are used almost never if a computing device with unlimited precision is utilized. Further, the asymptotic speed of convergence for any member is at least of order two. In view of the difficulties of obtaining the functional form of the second order partials of the likelihood function for general linear and nonlinear simultaneous systems, the method proposed here may be recommended in the computation of full information maximum likelihood estimates. Even if the partials of the system of equations are easily calculated, then some member of the family may still lead to convergence if the Newton-Raphson method does not. Practically speaking, the proposed method can be used to determine an approximate solution and this approximate solution will be closer to the solution if the precision of the computations is higher.