Maximum Likelihood Theory for Incomplete Data from an Exponential Family
Rolf Sundberg · 2016
In this paper we study such classes of distribu- tions which are generated from exponential families by loss of information due to the fact that only some function of the exponential family variable is observable. Examples of such classes are mixtures and convolutions of exponential type distributions as well as grouped, censored and folded distribu- tions. Their common structure is analysed. The existence is demonstrated of a n1/2-consistent, asymptotically normally distributed and asymptotically efficient root of the likelihood equation which asymptotically maximizes the likelihood in every compact subset of the parameter space, imposing only the natural requirement that the information matrix is positive definite. It is further shown that even the weaker requirement of local parameter identifiability, which admits of application to non-regular cases, is sufficient for the existence of con- sistent maximum likelihood estimates. Finally the subject of large sample tests based on maximum likelihood estimates is touched upon.