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A Complete Class of Type 1 Optimal Block Designs with Unequal Replications

Abstract

For block designs comparing v treatments in b incomplete blocks of size k, consider settings where bk = vr + 1 and \(r(k-1)=\lambda (v-1)\) for integers r and \(\lambda \). These settings admit designs that possess the symmetry of balanced incomplete block designs and which, though nonbinary, are candidates for optimality in some standard senses. For k = 3, earlier authors have established a class of one binary and one nonbinary design that is complete with respect to all type 1 optimality criteria. Here a solution for the complete class problem for type 1 optimality is obtained for k = 5. The complete class includes two binary and two nonbinary designs.

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Acknowledgements

This work began when the first author visited the Department of Statistics, Virginia Tech in May, 2016. She expresses her gratitude to everyone in the department for their hospitality during the visit. She also thanks Professor B.V. Rajarama Bhat, professor-in-charge, Stat-Math division, Indian Statistical Institute and Professor Abhyuday Mandal, Department of Statistics, University of Georgia for making this visit possible.

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Part of special issue guest edited by Pritam Ranjan and Min Yang—Algorithms, Analysis and Advanced Methodologies in the Design of Experiments.

Appendix: Results on Eigenvalues

Appendix: Results on Eigenvalues

Bounds involving the eigenvalues of a real symmetric matrix are provided here. In what follows, A will denote a real symmetric \(n \times n\) matrix with row sum zero. The rows and columns of A are indexed by \(N = \{1, \ldots n\}\). For \(I = \{i_1, \ldots , i_t\} \subset N\), the principal submatrix of A with rows and columns indexed by I will be denoted by A(I). In design optimality arguments, A will be either \(C_d\) or \(\Delta _d\), and n will be v.

The first result follows from Ky Fan’s Maximum principle (see page 24 of Bhatia [3], for instance).

Lemma 8.1

Consider \(n \times 1\) nonzero vectors \(x_i, i = 1, \ldots , p \; (p < n)\) satisfying \(x^{\prime }_i 1_n = 0 = x^{\prime }_i x_j,\, i \ne j, \; i,j =1, \ldots , p\). Then the following holds.

$$\begin{aligned} \sum _{j=1}^{i}\mu _j(A) \le \sum _{j=1}^{i}(x_j^{\prime }A x_j)/(x_j^{\prime }x_j) \le \sum _{j=1}^{i}\mu _{n-j} (A), 1 \le i \le p. \end{aligned}$$

Corollaries 8.1 and 8.2 are immediate consequences of Lemma 8.1.

Corollary 8.1

(a) Let \(I \subset N, \; |I| = m\) and \(A_1 = A(I)\). For \(m \times 1\) nonzero vectors \(x_i, \, i = 1, \ldots , p\) (\(p < m)\) satisfying \(x^{\prime }_i 1_m = 0 = x^{\prime }_i x_j,\) \(i\ne j =1, \ldots , p\),

$$\begin{aligned} \sum _{j=1}^{i}\mu _j(A) \le \sum _{j=1}^{i}(x_j^{\prime }A_1 x_j)/(x_j^{\prime }x_j) \le \sum _{j=1}^{i}\mu _{n-j} (A), \ 1 \le i \le p. \end{aligned}$$

(b) Let \(I_1\) and \(I_2\) be two disjoint subsets of N of sizes \(n_1\) and \(n_2\), respectively, and let \(A_t = A(I_t), \ t = 1,2\). Let \(x_{i1}, \ldots x_{im_i}\) be \(m_i\) \(n_i \times 1\) vectors satisfying \(x^{\prime }_{ij} 1_{n_i} = 0 = x^{\prime }_{ij} x_{il}, \ l \ne j,\; 1 \le l,j \le m_i, i = 1,2\). Consider integers \(p_1,p_2\) such that \(1 \le p_i \le m_i\). Let \(q = p_1 + p_2\). Then

$$\begin{aligned} \sum _{i=1}^{q}\mu _i(A) \le \sum _{t=1}^2 \sum _{j=1}^{p_i}(x_j^{\prime }A_t x_j)/(x_j^{\prime }x_j) \le \sum _{i=1}^{q} \mu _{n-i} (A), \ 2 \le q \le m_1 + m_2. \end{aligned}$$

Corollary 8.2

If A has a diagonal element p, then

$$\begin{aligned} \mu _1 (A) \le n p/(n-1) \le \mu _{n-1} (A). \end{aligned}$$

Lemma 8.2

If A has any one of the matrices \(M_1,\ldots ,M_6\) listed below as a principle submatrix, then \(\mu _1(A) \le -3/2\).

$$\begin{aligned} \begin{array}{lrlrlrl} M_1 & = & \left( \begin{array}{ll} 0 & 1^{\prime }_p \\ 1_p & 0_{p \times p} \end{array} \right) ,\,\, p \ge 3 & & M_2 & = & \left( \begin{array}{rrr} 0 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{array} \right) \\ & & & & & & \\ M_3 & = & \left( \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right) & & M_4 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array} \right) \\ & & & & & & \\ M_5 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array} \right) & & M_6 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \end{array} \right). \end{array} \end{aligned}$$

Proof

By Corollary 8.1(a) with \(p=1\), for each \(q \times q\) matrix \(M_i\) it is enough to find a \(q \times 1\) vector x such that \(x^{\prime }1_q = 0\) and \(x^{\prime }M_i x/(x^{\prime }x) \le -3/2 \). Here are the x’s and the upper bounds for \(\mu _1(M_i)\) they produce:

Submatrix

x-vector

Upper bound for \(\mu _1\)

\(M_1\)

\((p, -1^{\prime }_p)\)

\(-2p/(p+1)\)

\(M_2\)

(2, −1, −1)

−5/3

\(M_3\)

(1, −1, 1, −1)

−3/2

\(M_4\)

(1, 1, −1, −1)

−3/2

\(M_5\)

(1, −1, −1, 1)

−2

\(M_6\)

(1, 1, −1, −1)

−3/2

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Bagchi, S., Morgan, J.P. A Complete Class of Type 1 Optimal Block Designs with Unequal Replications. J Stat Theory Pract 13, 59 (2019). https://doi.org/10.1007/s42519-019-0057-4

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  • DOI: https://doi.org/10.1007/s42519-019-0057-4

Keywords

  • Block designs
  • Optimality
  • Type 1 criteria
  • Complete class