MONTE CARLO ESTIMATION OF THE SAMPLING DISTRIBUTION OF NONLINEAR PARAMETER ESTIMATORS

JAMES JOSEPH SWAIN, Purdue University

Abstract

The sampling distribution of parameter estimators arising from nonlinear estimation can be characterized by its moments, percentiles, or quantiles. The quantities can be approximated by power series, obtained through investigation of transformation of the parameters, or estimated directly by Monte Carlo. This research investigates a Monte Carlo control variate method based on a linear approximation of the parameter estimator. The control variate estimator is shown to be more efficient than direct Monte Carlo sampling over a range of nonlinear models, experimental conditions, and levels of nonlinearity. The control variate method can be applied to estimators of the mean and higher moments, covariances, percentiles, and quantiles. A generalization of the least squares linear approximator used as the control variate extend the method to regression estimators other than least squares, to nonnormal and correlated errors, and to non-explicit models. When the regression function is linear the efficiency of control variates relative to direct Monte Carlo sampling, is infinite. The relative efficiency decreases as the Beale (1960) measure of nonlinearity N(,(theta)) increases. However, the method is more efficient than direct Monte Carlo even for extreme nonlinearity. A simple function approximating the relative efficiency of control variates is E = (2N(,(theta)))('-p) where E is the relative efficiency, p is the dimension of the parameter estimator vector, and N(,(theta)) is Beale's measure of nonlinearity. The control variate used in this research is the standard linear approximator for (')(theta), (delta) = (F('T)F)('-1) F (epsilon) where F is the n x p Jacobian matrix of the nonlinear regression function evaluated at the true values of the parameters, (theta)(,0), and (epsilon) is the n vector of errors. In certain cases the linear approximation of a transformation of (theta) (e.g., (phi)(,k) = ((theta))('k)) is a more advantageous control variate.

Degree

Ph.D.

Subject Area

Statistics

COinS