Tests for global maximum of the likelihood function

Given a relative maximum of the log-likelihood function, how to assess whether it is the global maximum? This paper investigates a statistical tool, which answers this question by posing it as a hypothesis testing problem. A general framework for constructing tests for the global maximum is given. The characteristics of the tests are investigated for two cases: correctly specified model and model mismatch. A finite sample approximation to the power is given, which gives a tool for performance prediction and a measure for comparison between tests. The tests are illustrated for two applications: estimating the parameters of a Gaussian mixture model and direction finding using an array of sensors - practical problems that are known to suffer from local maxima.