Non-Regular Maximum Likelihood Problems
本文回顾并讨论了四种非正则估计问题，包括无界似然问题和参数位于特殊子区域时的三种情况，并展示了如何系统处理这些问题。
SUMMARY Four non-regular estimation problems are reviewed and discussed. One (the unbounded likelihood problem) involves distributions with infinite spikes, for which maximum likelihood can fail to give consistent estimators. A comparison is made with modified likelihood and spacings methods which do give efficient estimators in this case. An application to the Box–Cox shifted power transform is given. The other three problems occur when the true parameter lies in some special subregion. In one (the constrained parameter problem) the subregion is a boundary. The other two (the embedded model and the indeterminate parameters problems) occur when the model takes on a special form in the subregion. These last two problems have previously been investigated separately. We show that they are equivalent in some situations. Both often arise in non-linear models and we give a directed graph approach which allows for their occurrence in nested model building. It is argued that many non-regular problems can be handled systematically without having to resort to elaborate technical assumptions. Relatively uncomplicated methods may be used provided that the underlying nature of the non-regularity is understood.