The Expected Sample Variance of Uncorrelated Random Variables with a Common Mean and Some Applications in Unbalanced Random Effects Models

Abstract

There is a little-known but very simple generalization of the standard result that for uncorrelated random variables with common mean and variance , the expected value of the sample variance is . The generalization justifies the use of the usual standard error of the sample mean in possibly heteroscedastic situations, and motivates elementary estimators in even unbalanced linear random effects models. The latter both provides nontrivial examples and exercises concerning method-of-moments estimation, and also helps “demystify” the whole matter of variance component estimation. This is illustrated in general for the simple one-way context and for a specific unbalanced two-factor hierarchical data structure.

• Heteroscedastic
• Method of moments
• One-way model
• Two-factor hierarchical model
• Standard error of the mean; Variance component

Acknowledgments

The authors thank Jim Stapleton for pointing out the reference to his text, and the fact that the proof of Lemma 1 can be simplified by noting that without loss of generality one may assume that . They also thank Joachim Kunert, Ron Christensen, Bobby Mee, Karen Kafadar, Glen Meeden, H.A. David, Dennis Gilliland, Götz Trenkler, Bob Stephenson, and three anonymous reviewers for comments on earlier drafts of the note that have worked to make it more complete and readable, and hopefully more useful.

Financial support of the Deutsche Forschungsgemeinschaft (SFB 475, “Reduction of Complexity in Multivariate Data Structures”) through the University of Dortmund and of the Los Alamos National Laboratory Statistical Sciences Group is gratefully acknowledged by the first author.