Three Papers on Maximum Likelihood in Exponential Families

General methods for obtaining maximum likelihood estimates in exponential families are demonstrated using a constrained autologistic model for estimating relatedness from DNA fingerprint data. The novel features are the use of constrained optimization and two new algorithms for maximum likelihood estimation. The first, the "phase I" algorithm determines the support of the MLE in the closure of the exponential family (a distribution in the family conditioned on a face of the convex support of the natural statistic) when the MLE does not exist in the traditional sense (a point in the natural parameter space). The second, the maximum Monte Carlo likelihood algorithm uses the Metropolis algorithm or the Gibbs sampler to obtain estimates when exact calculation of the likelihood is not possible. Separate papers on each algorithm accompany the paper illustrating a complete analysis using these new methods.