J. Appl. Math. & Computing Vol. 16(2004), No. 1 - 2, pp. 143 - 150
A-OPTIMAL CHEMICAL BALANCE WEIGHING DESIGN WITH CORRELATED ERRORS BRONISLAW CERANKA∗ AND MALGORZATA GRACZYK
Abstract. In this paper we study the estimation problem of individual weights of objects using an A-optimal chemical balance weighing design. We assume that in this model errors are correlated and they have the same 0 variances. The lower bound of tr(X G−1 X)−1 is obtained and a necessary and sufficient condition for this lower bound to be attained is given. There is given new construction method of A-optimal chemical balance weighing design. AMS Mathematics Subject Classification: 62K15. Key words and phrases : A-optimal chemical balance weighing design.
1. Introduction We consider the model of the chemical balance weighing design y = Xw + e,
(1)
where the n × p design matrix X has elements equal to -1, 0 or 1 if the one from p objects is kept on the left pan, right pan or is not included in the particularly weighing, respectively. y is an n × 1 random observed vector of the recorded results of weights, w is a p×1 column vector representing the unknown weights of objects and e is an n×1 random vector of errors.0 We assume that E(e) = 0n and V ar(e) = σ 2 G, where G = g (1 − ρ)In + ρ1n 1n , g > 0, −1 < ρ < 1 are given constants. In other words, errors are correlated and they have the same variances 0 and there are not systematic errors. Let us notice, that G = g (1−ρ)In +ρ1n 1n is positive definite matrix if and only if −1 < ρ < 1. n−1 Received May 30, 2003. Revised January 17, 2004. ∗ Corresponding author. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.
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Bronislaw Ceranka, Malgorzata Graczyk
For estimation of individual unknown weights of objects we use the normal equations 0
0
X G−1 Xw ˆ = X G−1 y,
(2)
where w ˆ is the vector of the weights estimated by the least squares method. The chemical balance weighing design is singular or nonsingular depending 0 on whether the matrix X G−1 X is singular or nonsingular, respectively. It is obvious that because of the assumption connected with the matrix G the matrix 0 0 X G−1 X is nonsingular if and only if the matrix X X is nonsingular, i.e. if and 0 only if X is of full column rank (= p). If X G−1 X is nonsingular, the least squares estimator of w is given in the form −1 0 −1 0 w ˆ = X G−1 X XG y (3) and the variance - covariance matrix of w ˆ is given by the formula −1 0 V ar(w) ˆ = σ 2 X G−1 X .
(4)
In this paper we consider the problem of choosing the matrix X under which −1 0 the sum of variances of w ˆ1 , w ˆ2 , ..., w ˆp , i.e. σ 2 tr X G−1 X is minimum when 0 G = g (1 − ρ)In + ρ1n 1n is the positive definite matrix. This design is called A-optimal design. In the case G = In , several methods of the construction of the A-optimal chemical balance weighing designs are given in literature (see [1], [7], [8]). In the paper [2] we considered those designs when G is the positive definite diagonal matrix. For further discussion of an A- and the other optimality criteria see [3], [4], [6]. 0
2. A lower bound for tr X G−1 X
−1
In many issues concerned on weighing problems we consider the A-optimal −1 0 designs. There are designs in that tr X G−1 X attain the lower bound. Let X be an n × p design matrix of a chemical balance weighing design and 0 −1 G = g (1 − ρ)In + ρ1n 1n , g > 0, 8. Proof. The proof of this theorem is similar to the proof of Theorem 5. 0
If the condition xi xj = n − 8, i 6= j, i, j = 1, 2, ..., p is fulfilled then from (ii) we have Corollary 3. If (i) n = 5, (ii) n = 6, (iii) n = 7,
−1 n−1
< ρ < 0, m = 2 and
3 , ρ = − 13 1 ρ = −7, 1 , ρ = − 15
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Bronislaw Ceranka, Malgorzata Graczyk
then the A-optimal chemical balance weighing design with the design matrix X and with the variance-covariance matrix of errors σ 2 G, where G is given by (5), exists. 3 Example. Let n = 5, m = 2, ρ = − 13 , p = 2, g = 1. Then 0 16 G−1 = I5 + 315 15 13 and −1 1 −1 1 1 −1 X= 1 −1 1 −1 is the design matrix of the A-optimal chemical balance weighing design with the above variance-covariance matrix of errors.
References 1. K. S. Banerjee, Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics. Marcel Dekker Inc., New York, 1975. 2. B. Ceranka and K. Katulska, A-optimal chemical balance weighing designs with diagonal covariance matrix of errors, MODA 6, A. C. Atkinson, P. Hackl and W. G. Mller, eds., Physica, Heidelberg, 2001. 3. C. S. Cheng, Optimality of certain assymetrical experimental designs. Ann. Statist. 6 (1978), 1239-1261. 4. Z. Gail and J. Kiefer, D-optimum weighing designs. Ann. Statist. 8 (1980), 1293-1306. 5. H. Hotelling, Some improvements in weighing designs and other experimental techniques, Ann. Math. Stat. 15 (1944), 297-305. 6. J. Kiefer, General equivalence theory for optimum design (approximate theory). Ann. Statist. 2 (1974), 849-874. 7. D. Raghavarao, Constructions and Combinatorial Problems in Designs of Experiments. John Wiley Inc., New York, 1971. 8. S. C. Wong and J. C. Masaro, A - optimal matrices X = (xij )N×n with xij = −1, 0, 1, Linear and Multilinear Algebra 15 (1984), 23-46. Bronislaw Ceranka, received his doctor title in 1972 from the Adam Mickiewicz University in Pozna´ n, his professor title in 1991 from the Agricultural University of Pozna´ n, Poland. His research interests are the experimental designs and applicactions of them. Also he does statistical consulting professor. Malgorzata Graczyk, received her doctor title in 2002 from the Adam Mickiewicz University in Pozna´ n, Poland Department of Mathematical and Statistical Methods, Agricultural University of Pozna´ n, ul. Wojska Polskiego 28, 60-637 Pozna´ n, Poland e-mail:
[email protected],
[email protected] 
