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.