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    188宝金博页面版: Determinant formulas with applications to designing when the observations are correlated

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    内容提示： Ann. Inst. Statist. Math. Vol. 47, No. 2, 385-399 (1995) DETERMINANT DESIGNING WHEN FORMULAS WITH APPLICATIONS TO THE OBSERVATIONS ARE CORRELATED* WOLFGANG BISCHOFF Institute of Mathematical Stochastics, Department of Mathematics, University of Karlsruhe, D-76128 Karlsruhe, Germany (Received June 13, 1994; revised December 5, 1994) Abstract. In the general linear model consider the designing problem for the Gaufi-Markov estimator or for the least squares estimator when the ob- servations are correlated. De...

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    Ann. Inst. Statist. Math. Vol. 47, No. 2, 385-399 (1995) DETERMINANT DESIGNING WHEN FORMULAS WITH APPLICATIONS TO THE OBSERVATIONS ARE CORRELATED* WOLFGANG BISCHOFF Institute of Mathematical Stochastics, Department of Mathematics, University of Karlsruhe, D-76128 Karlsruhe, Germany (Received June 13, 1994; revised December 5, 1994) Abstract. In the general linear model consider the designing problem for the Gaufi-Markov estimator or for the least squares estimator when the ob- servations are correlated. Determinant formulas are proved being useful for the D-criterion. They allow, for example, a (nearly) elementary proof and a generalization of recent results for an important linear model with multiple re- sponse. In the second part of the paper the determinant formulas are used for deriving lower bounds for the efficiency of a design. These bounds are applied in examples for tridiagonM covariance matrices. For these examples maximin designs are determined. Key words and phrases: Determinant formula, general linear model, corre- lated observations, D-criterion, efficiency of designs, linear model with multi- ple response, lower bounds for the efficiency, tridiagonal matrices as covariance structure, maximin designs. 1. Introduction, notations and preliminary results Consider a general linear model Y=X/3+ z where X is a known real (n × m)-matrix, n _ m,/3 E ~m is an unknown parameter vector and Z is an n-dimensional real random vector with EZ = 0n = (0,..., 0) T E ~, Cov Z = C positive definite. In this paper we are interested in estimating/3. In the first instance we assume that C is known. For/3 being estimable X must be of full rank, that is, rank(X) = m. We consider two estimators mostly used for estimating/3. Firstly the best linear unbiased estimator, the so-called Gaufi-Markov estimator (xTc-1x)-IxTc-1 : ~n ~ ~m * Parts of the paper are based on a part of the author's Habilitationsschrift Bischoff (1993a). 385

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