Testing for Uniqueness of Estimators

Uniqueness of the population value of an estimated descriptor is a standard assumption in asymptotic theory. However, m-estimation problems often allow for local minima of the sample estimating function, which may stem from multiple global minima of the underlying population estimating function. In the present article, we provide tools to systematically determine for a given sample whether the underlying population estimating function may have multiple global minima. To achieve this goal, we develop asymptotic theory for non-unique minimizers and introduce asymptotic tests using the bootstrap. We discuss three applications of our tests to data, each of which presents a typical scenario in which non-uniqueness of descriptors may occur. These model scenarios are the mean on a non-euclidean space, non-linear regression and Gaussian mixture clustering.