Multiple roots of estimating functions

Estimating functions can have multiple roots. In such cases, the statistician must choose among the roots to estimate the parameter. Standard asymptotic theory shows that in a wide variety of cases, there exists a unique consistent root, and that this root will lie asymptotically close to other consistent (possibly inefficient) estimators for the parameter. For this reason, attention has largely focused on the problem of selecting this root and determining its approximate asymptotic distribution. In this paper, however, we concentrate on the exact distribution of the roots as a random set. In particular, we propose the use of higher-order root intensity functions as a tool for examining the properties of the roots and determining their most problematic features. The use of root intensity functions of first and second order is illustrated by application to the score function for the Cauchy location model.