Counter-examples design to global convergence of maximum likelihood estimators

MLE(Maximum Likelihood Estimation) is widely applied in system identification because of its consistency, asymptotic efficiency and sufficiency. However gradient-based optimization of the likelihood function might end up in local convergence. To overcome this difficulty, the non-local-minimum conditions are very useful. Here we suggest a heuristic method of constructing local minimum examples for ARMAX, ARARMAX and BJ models. Based on them the derivation of non-local-minimum conditions can be inspired by analyzing these examples.