Higher order statistical properties of nonlinear estimators.

摘要

Certain higher order statistical properties of nonlinear estimators are derived and analyzed. The third order bias of these estimators is derived using stochastic expansions. The higher order terms are strongly influenced by the higher order derivatives of the model being estimated. Also, the higher order terms in the expansion depend critically on moments which may or may not exist for quite common distributions. It may be prudent to allow for these moments when conducting inferences, particularly when using small to moderate sized samples.;The results are shown to apply to most of the common extremum estimators used in applied work including GMM, MLE, and Generalized Empirical Likelihood estimators in an i.i.d. sampling context. Examples of the applicabilty of the results are provided for a number of popular nonlinear estimators such as the nonlinear simultaneous equation model, nonlinear regression models and the exponential regression model.;The viability of the results are shown using a Monte Carlo experiment for the exponential regression model. In this experiment, higher order analytical bias corrections to the estimators are shown to substantially reduce bias in comparison to higher order bootstrap and jackknife corrections. This is especially so in smaller samples. Also, the performance of confidence intervals constructed using an Edgeworth approximation for the estimators is found to be superior to various types of bootstrap confidence intervals. The methods developed are therefore useful for improved inferences in finite samples.;An asymptotic approximation of the distribution for nonlinear estimators is derived using an Edgeworth expansion. A simple method is developed using a linear transformation of a vector of parameters. This is done for both studentized and non-studentized statistics. The analytical corrections provided by the higher order terms in the Edgeworth expansions provide an intuitive explanation for the departure of the finite sample distribution of nonlinear estimators from the asymptotic distribution.