For small sample random effects models, results are derived which show in certain cases, and indicate in general, that an estimated random effects variance matrix may be used in the weight matrices without causing undue error in the empirically weighted mean. Exact error variances are derived mathematically for the empirically weighted mean for the two sample case in one and two dimensions. Simulation is used to determine errors for a practical example of six five-variate samples. For estimation of their mean, the differences between the samples are ancillary. The biases of the average and weighted mean estimators conditional on these ancillaries is illustrated in a diagram plotting values obtained by simulation. A curious range anomaly is illustrated which arises if random effects are ignored when present.
"Matrix Weighting of Several Regression Coefficient Vectors." Ann. Statist. 21 (2) 1093 - 1114, June, 1993. https://doi.org/10.1214/aos/1176349166