Eliminating multiple root problems in estimation (with comments by John J. Hanfelt, C. C. Heyde and Bing Li, and a rejoinder by the authors)

Abstract

Estimating functions, such as the score or quasiscore,can have more than one root. In many of these cases, theory tells us that there is a unique consistent root of the estimating function. However, in practice, there may be considerable doubt as to which root is appropriate as a parameter estimate. The problem is of practical importance to data analysts and theoretically challenging as well. In this paper, we review the literature on this problem. A variety of examples are provided to illustrate the diversity of situations in which multiple roots can arise. Some methods are suggested to investigate the possibility of multiple roots, search for all roots and compute the distributions of the roots. Various approaches are discussed for selecting among the roots. These methods include (1) iterating from consistent estimators, (2) examining the asymptotics when explicit formulas for roots are available, (3) testing the consistency of each root, (4) selecting by bootstrapping and (5) using information-theoretic methods for certain parametric models. As an alternative approach to the problem, we consider how an estimating function can be modified to reduce the number of roots. Finally, we survey some techniques of artificial likelihoods for semiparametric models and discuss their relationship to the multiple root problem.