Exact Distribution of the Nonlinear Least Squares Estimator

• Activity Number: 345
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Eugene Demidenko*+
• Companies: Dartmouth Medical School
• Keywords: estimating equation ; small sample ; exact confidence level ; nonlinear regression ; saddle point approximation

Abstract

The exact density distribution of the Nonlinear Least Squares (NLS) estimator is derived in closed form and expressed through the cumulative distribution function of the standard normal variable for a general univariate nonlinear regression model in small samples. The exact density is extended to the estimating equation (EE) approach and nonlinear regression with an arbitrary number of linear parameters and a coefficient at the nonlinear function (partial linear least squares). For a very special nonlinear regression model, the exact density of the NLS estimator coincides with the distribution of the ratio of two normally distributed random variables derived previously by Fieller (1932). Approximations to the density of the NLS and EE estimators are developed in the multivariate case as well. Numerical issues are illustrated, such as nonexistence and multiple solutions of the normal equation as the major factors of poor density approximation. Density computations and simulation results are reported for several popular examples of nonlinear regression models, including the Michaelis-Menten model.