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The Variance of a Linear Combination of Independent Estimators Using Estimated Weights NIE

Rubin, Donald B.; Weisberg, Sanford.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
National Institute of Education (NIE), Computer Software, Mathematical Formulas


Let t(subl),...,t(sub K) be independent unbiased estimators of a parameter tau. Let t* = SUM alpha(sub i)t(sub i) have minimum variance among unbiased linear estimators of tau, and let t(hat) = SUM alpha(hat, sub i)t(sub i, SUM alpha(hat, sub i) = 1, be any linear combination with (alpha(hat,sub 1),..., alpha(hat, sub k)) independent of t(sub 1),...,t(sub k). We prove that t(hat) is unbiased for tau and that the ratio of the variance of t(hat) to the variance of t* is 1 + SUM MSE (alpha(hat, sub i))/alpha(sub i) where MSE (alpha(hat,sub i)) is the mean square error of alpha(hat, sub i) as an estimator of alpha(sub i). (7pp.)

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