Combining estimators of a common parameter across samples

In many settings, multiple data collections and analyses on the same topic are summarised separately through statistical estimators of parameters and variances, and yet there are scientific reasons for sharing some statistical parameters across these different studies. This paper summarises what is known from large-sample theory about when estimators of a common structural parameter from several independent samples can be combined functionally, or more specifically linearly, to obtain an asymptotically efficient estimator from the combined sample. The main idea is that such combination can be done when the separate-sample nuisance parameters, if any exist, vary freely and independently of one another. The issues are illustrated using data from a multi-centre lung cancer clinical trial. Examples are presented to show that separate estimators cannot always be combined in this way, and that the functionally combined separate estimators may have low or 0 efficiency compared to the unified analysis that could be performed by pooling the datasets.