Asymptotic Distribution of Log-Likelihood Maximization Based Algorithms and Applications

The asymptotic distribution of estimates that are based on a sub-optimal search for the maximum of the log-likelihood function is considered. In particular, estimation schemes that are based on a twostage approach, in which an initial estimate is used as the starting point of a subsequent iterative search, are analyzed. The analysis is relevant for cases where the log-likelihood function is known to have local maxima in addition to the global maximum, and there is no available method that is guaranteed to provide an estimate within the attraction region of the global maximum. In addition, an algorithm for finding the maximum likelihood estimator is offered. The algorithm is best suited for scenarios in which the likelihood equations do not have a closed form solution, the iterative search is computationally cumbersome and highly dependent on the data length, and there is a risk of convergence to a local maximum. The result on the asymptotic distribution is validated and the performance of the offered algorithm is examined by computer simulations.