MLE, Information, Ancillary Complement, and Conditional Inference with Illustrations

Abstract

Fisher’s (Proceedings of Royal Society Series A 144, 285–307 1934, 1956) example remains a classic where the maximum likelihood estimator (T) was non-sufficient, had less than full information, but an ancillarity complement (S) helped in recovering the full information $\mathcal {I}_{(T,S)}(\theta )$. In the absence of other readily accessible easy-to-grasp examples of similar nature, we begin with general calculations for useful information entities, both unconditional ($\mathcal {I}_{T}(\theta )$) and conditional ($\mathcal {I}_{T\mid S}(\theta )$). These have led us to propose a number of new illustrations in the spirit of the original example. Then, we introduce a multivariate data extension of the original example with an illustration. We wrap up this investigation with an example of a non-sufficient MLE T that has (i) the full Fisher information, and (ii) has an ancillary complement S.