Nonlinear Estimation and Asymptotic Approximations - Ramsey, James B. (Econometrica, 1978)
central objective of this paper is to present a series expansion of nonlinear estimators in order to facilitate an analysis of the distributions of such estimators. Where the estimator under consideration is a maximum likelihood estimator, the method provides somewhat more information, as well as higher order approximations to the distributions of the nonlinear estimators than does the usual theory which demonstrates asymptotic normality. The method is also useful for a wide class of estimators including those defined only implicitly by the estimating procedure. Approximations to the distributions of the nonlinear estimators can be obtained in many cases even when the moments do not exist. In any event, it is to be hoped that the analytic procedures discussed in this paper will simplify the analysis of specific cases and will shed more light on the general formulation of nonlinear estimation problems. The remainder of this paper is in four sections. The first section presents the basic theory and analyzes the asymptotic distributions of nonlinear estimators in correctly specified models. This is followed in the second section by a brief discussion of a number of interesting examples. The third section compares the approach outlined in this paper with the traditional maximum likelihood and general nonlinear series expansions. In the fourth section the approximate asymptotic distribution of the regression residuals is derived. The general statement of the model to be considered in the following sections is given by: