Nuisance parameters, mixture models, and the efficiency of partial likelihood estimators

Abstract This paper establishes lower bounds for estimation in parametric statistical models in which one wishes to estimate a real-valued parameter of interest in the presence of nuisance parameters which are accruing in number in direct proportion to the number of independent observations. The formal setting requires that the nuisance parameters be independent observations from an unknown distribution. In this setting an information measure analogous to the Fisher information is derived. It is then used to generate lower bounds for the variance of unbiased estimators and also for the asymptotic variance of consistent asymptotically normal estimators. Under certain conditions, consistent asymptotically normal estimators can be generated by maximizing factors of the complete likelihood, even though the maximum likelihood estimator is inconsistent. These estimators can be fully efficient in the sense of meeting the lower bounds despite their apparent wasteful use of the likelihood, as is demonstrated, in several important examples, by the use of a natural sufficient condition.