Multiple roots of estimating functions and applications

Abstract

An estimating function may give multiple solutions. This creates a certain amount of confusion as to which of these roots is the most appropriate choice as an estimate of parameter. This thesis discusses the existing methods, and proposes two new approaches to choose the best root among multiple roots. One approach is based on the root intensity, which is an extension of the probability density function of an estimator to the multiple root case. This method is also applied to some practical examples such as logistic regression models with measurement error and bivariate normal mixture models. Another one is the shifted information method, which can be used in transformation models and, in particular, the location models. Though multiple roots of estimating functions arise in some cases, it can be shown that under some regularity conditions, there is a unique root with high probability in a given bounded closed set which includes the true value.