On Tests for Global Maximum of the Log-Likelihood Function

D. Blatt, A.O. Hero
2007 IEEE Transactions on Information Theory  
Given the location of 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 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
more » ... 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 distance between the true underlying distribution and the assumed parametric class and tests that are insensitive to small deviations from the model are derived. 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. Index Terms-Parameter estimation, maximum likelihood, global optimization, local maxima, array processing, Gaussian mixtures, superimposed exponentials in noise.
doi:10.1109/tit.2007.899537 fatcat:22ibnyoi55h5xgng7pyttxk7ne