Testing for a Global Maximum of the Likelihood

Abstract

When several roots to the likelihood equation exist, the root corresponding to the global maximizer of the likelihood is generally retained. This procedure, however, supposes that all possible roots are identified. Because in many cases the global maximizer is the only consistent root, we propose a test to detect if a given solution is consistent. This test relies on some necessary and sufficient conditions for consistency of a root and simply consists of comparing the difference between two expected log-likelihood expressions. Monte Carlo studies and a real life example show that the proposed procedure leads to encouraging results. In particular, it clearly outperforms another available test of this kind, especially for relatively small sample sizes.