Solving Non-Linear Estimation Equations

Abstract

SUMMARY In this paper we consider the numerical computation of a vector parameter estimate θ^ which is a root of a system of unbiased non-linear estimation equations. A sequence {θ (r)} is constructed which converges with a probability approaching 1 to θ^ from any starting value θ (0). Unlike Newton-type algorithms, the θ (r) do not involve matrices of partial derivatives. Furthermore it can be shown that θ(r)−θ^ is of the order Op(n – r/2) so that great accuracy is achieved in very few iterations. When the θ (r) do not have closed forms, an approximation is suggested. An alternative and usually simpler way is also found for deriving analytically the expected value of the matrix of partial derivatives in the asymptotic covariance matrix of θ^. The efficiency of the proposed algorithm is demonstrated numerically and compared with that of the EM and the Newton–Raphson algorithms. Applications to some well-known statistical problems, including missing data analysis, variance component models, familial correlations and errors-in-variables models are also briefly discussed.