On tests for global maximum of the log-likelihood function

Given the location of a relative maximum of the log-likelihood function, how to assess whether it is the global maximum? This paper investigates an existing statistical tool, which, based on asymptotic analysis, answers this question by posing it as a hypothesis testing problem. A general framework for constructing tests for 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 sensitivity of the tests to model mismatch is analyzed in terms of the Renyi divergence and the Kullback-Leibler divergence between the true underlying distribution and the assumed parametric class and tests that are insensitive to small deviations from the model are derived thereby overcoming a fundamental weakness of existing tests. The tests are illustrated for three applications: passive localization or direction finding using an array of sensors, estimating the parameters of a Gaussian mixture model, and estimation of superimposed exponentials in noise-problems that are known to suffer from local maxima.