Estimation of a Common Mean and Weighted Means Statistics

Measurements made by several laboratories may exhibit non-negligible between-laboratory variability, as well as different within-laboratory variances. Also, the number of measurements made at each laboratory often differ. A question of fundamental importance in the analysis of such data is how to form a best consensus mean, and what uncertainty to attach to this estimate. An estimation equation approach due to Mandel and Paule is often used at the National Institute of Standards and Technology (NIST), particularly when certifying standard reference materials. Primary goals of this work are to study the theoretical properties of this method, and to compare it with some alternative methods, in particular to the maximum likelihood estimator. Towards this end, we show that the Mandel-Paule solution can be interpreted as a simplified version of the maximum likelihood method. A class of weighted means statistics is investigated for situations where the number of laboratories is large. This c...