D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon_{i},\ i=1,2,\dots,n,}$$\end{document} are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{\bf X}}$$\end{document} and we prove that this design is D-optimal in certain classes of designs for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho\in[0,1)}$$\end{document} and it is also D-optimal in the class of designs with the design matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf X} \in M_{n\times 3}(\pm 1)}$$\end{document} for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{n\times 3}(\pm 1)}$$\end{document} , if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho\in[0,1/(n-2))}$$\end{document} . We present also the matrix of the D-optimal factorial design with 3 two-level factors.


Definition 1
The design with the design matrix X D is D-optimal in the class of designs with the design matrix in the set C ⊆ M n×3 (±1) if det(X D S −1 X D ) = max det(X S −1 X) : X ∈ C .
The matrix 1 1−ρ 2 A is the inverse matrix of S, where The matrix A is positive definite for ρ ∈ (−1, 1).

Lemma 1 Let A be given by (2). The design X D is D-optimal in the class of designs with design matrices in C ⊆ M n×3 (±1) if
det(X D AX D ) = max det(X AX) : X ∈ C .
The case ρ = 0 is well known in the literature. The D-optimality problem is considered for example in Hadamard (1893), Cohn (1967Cohn ( , 1989, Galil andKiefer (1980), Cheng (1980) or Jacroux et al. (1983). For ρ = 0, the problem of the D-optimal design was considered in Li and Yang (2005), Yeh and Lo Huang (2005) and Katulska and Smaga (2010). In these papers, the D-optimality problem was solved in some subclasses of M n×3 (±1). We show in Sect. 2 that the design with the design matrix and k = 1, 2, . . ., is D-optimal in certain classes of designs for ρ ∈ [0, 1) and this design is D-optimal in the class of designs with design matrices X ∈ M n×3 (±1) for ρ ∈ [0, 1/(n − 2)]. We present also theorems giving necessary and sufficient conditions under which the design for three objects is D-optimal. At the end of Sect. 2, we compare our results with results in Yeh and Lo Huang (2005). In Sect. 3, we present the matrix of the D-optimal factorial design with 3 two-level factors, which is constructed by the matrix of D-optimal weighing design for three objects. In the appendix, we give some lemmas and theorems, which contain properties used in proofs of main results in this paper, and present the proofs of theorems from Sect. 2 and 3.
Theorem 1 If n ≡ 0 (mod 4) and ρ ∈ [0, 1), then the design with the design matrix X given by (3) is D-optimal in the class of designs with design matrices The proof of Theorem 1 is given in the "Appendix". In the following theorem, we consider the whole class of designs M n×3 (±1) and we show that the design X of the form (3) is still D-optimal in this class if ρ ∈ [0, 1/(n − 2)].
The proof of Theorem 2 is given in the "Appendix".
We now formulate the necessary and sufficient conditions under which the design is D-optimal in M n×3 (±1).
The proof of Theorem 3 is given in the "Appendix".
The proof of Theorem 4 is given in the "Appendix". Yeh and Lo Huang (2005) considered the exact D-optimal designs with 2 two-level factors and n autocorrelated observations. They proved that the design with the design matrixX , is D-optimal in considered class of designs for ρ ∈ (0, 1). But as it is described in Banerjee (1975), the design from Yeh and Lo Huang (2005) can be used as the design for finding the weights of three objects. It is easy to calculate that so the designX given by (5) is D-worse than the design X given by (3). Therefore, the D-optimal design from the subclass of designs with the first column of all ones is not D-optimal in the set M n×3 (±1).

D-optimal 2 factorial design
In this section, we consider designs of 3 factorial main effects with a first-order autoregressive errors. We describe the model similarly to Cheng (1980). Suppose we have 3 factors each with two levels 0 and 1. For any observation y j 1 j 2 j 3 on the j i th level of factor i, i = 1, 2, 3, j i = 0 or 1, E(y j 1 j 2 j 3 ) can be written as where φ 0 (grand mean) and φ 1 i , i = 1, 2, 3 (main effects) are unknown parameters and The sequence i of error terms is a first-order autoregressive process. In this setting, the design matrix X ∈ M n×4 (±1) has the first column of all ones. Then D-optimality criterion is defined similarly to Definition 1. In this model, we proved the following theorem.
Theorem 5 Let n ≡ 0 (mod 4) and ρ ∈ (0, 1/(n − 2)). If X * is the matrix of the D-optimal weighing design for three objects, then the design with the matrix [1, X * ] is D-optimal factorial design with 3 two-level factors with a first-order autoregressive errors.
The proof of Theorem 5 is very long and similar to the proofs of Theorems 1, 2 and 3, so we present only the sketch of it in the "Appendix".
Acknowledgments Authors are thankful to the referees for their constructive suggestions which improved the presentation of this article.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Appendix
First, we give several well known results.
Lemma 2 Let X be an n × m matrix and B be an n × n positive definite matrix. Then matrix X BX is positive semi-definite and rank X BX = rank (X). Moreover, X BX is positive definite if and only if a matrix X is of full column rank. Now, we present some simple lemmas.
Lemma 3 Let X ∈ M n×3 (±1). Then the determinant of the matrix X AX, where A is given by (2), does not change if we interchange two columns of the matrix X or we multiply any column of this matrix by −1.
Proof It is easy to see that Proof This is clear because the function g is the quadratic function with zero in t 0 = −2 1+ρ for fixed ρ ∈ (0, 1). Now, we present proofs of the theorems.
Proof of Theorem 1 If ρ = 0, then the matrix S given by (1) is equal to the identity matrix and from Hadamard (1893) we have det( X S −1 X) = det( X X) ≥ det(X X) = det(X S −1 X), because X X = nI, where I is the identity matrix of size 3. Therefore, we assume that ρ ∈ (0, 1). From Lemma 3, we suppose that the first row of the matrix X is [1, 1, 1]. The matrix X AX is positive definite from Lemma 2, because the matrix A given by (2) is positive definite and rank(X) = 3. We must consider three cases. We show in Case i, i = 1, 2, 3, that det( X A X) ≥ det(X AX) for all X ∈ C i , and by Lemma 1, we conclude that the matrix X given by (3) is the matrix of the D-optimal design in the considered class of the designs.

Case 3 X = [x, y, z] ∈ C 3
We can write C 3 as C 3 = C 3.1 ∪ C 3.2 ∪ C 3.3 , where In this case, the proof follows from the following subcases.
Proof of Theorem 2 The D-optimality of the design with the design matrix X given by (3), when ρ = 0, follows by the same argument as in the beginning of the proof of Theorem 1. So, we assume that ρ ∈ (0, 1/(n − 2)] and (by Lemma 3) the first row of X contains all ones. Therefore, we can consider the class of the design matrices C 1 ∪ C 2 ∪ C 3 ∪ C 4 , where C 1 , C 2 , C 3 are described in Theorem 1 and The proof is similar to that in Theorem 1. From that theorem, it follows that det( X A X) ≥ det(X AX) if X = [x, y, z] ∈ C 1 ∪C 2 ∪C 3 . Assume that X = [x, y, z] ∈ C 4 . Without loss of generality we can suppose that fcons(y) = a > b = fcons(z). From Lemmas 5(a), (c) and 4, we obtain We must divide the proof into two cases.
Case 1 If a = n 2 , then from Fischer's inequality, it follows that Case 2  and from Fischer's inequality, we conclude that According to Case 1 we have det( X A X) ≥ det(X AX) and the proof is complete.
(⇐) It is clear, because det(X * AX * ) = det( X A X), where X is given by (3) and it is a D-optimal design by Theorem 2.
Proof of Theorem 4 (⇐) By Lemmas 4 and 5, the matrix X * AX * has the form (4). Therefore, from Theorem 3 the design X * is D-optimal.
The proof is complete.