## Electronic Theses and Dissertations

1983

Doctoral Thesis

Ph.D.

#### Department

Mathematics and Statistics

Mathematics.

#### Rights

info:eu-repo/semantics/openAccess

#### Abstract

Net N and n be positive integers with N (GREATERTHEQ) n and let D(N,n) denote the set of all N x n matrices X = (x(,ij)) with x(,ij) = -1,0 or 1. Let D'(N,n) be the set of all matrices in D(N,n) with entries -1,1. Each such matrix in D(N,n) or D'(N,n) will be called a weighing design matrix. If X(,0) minimizes (PHI)(X('T)X) over D(N,n) for some real valued function (PHI), then X(,0) is said to be (PHI)-optimum over D(N,n). The characterization of such X(,0) arises from the statistical problems of weighing designs, certain block designs and 2('n) fractional factorial designs. The well-known D-, A- and E-optimality criteria are obtained by taking (PHI)(X('T)X) = det(X('T)X)('-1), tr(X('T)X)('-1) and the maximum of the eigenvalue of (X('T)X)('-1) respectively. All these criteria are functions of the spectrum of X('T)X. Let (lamda)(,1),(lamda)(,2),...,(lamda)(,n) be the eigenvalues of X('T)X. Then a more general family of criteria are the following (PHI)(,p)-criteria: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The A-criterion is the (PHI)(,1)-criterion, the E-criterion is the limit of the (PHI)(,p)-criterion as p (--->) (INFIN) and the D-criterion is the limit of the (PHI)(,p)-criterion as p (--->) 0('+). In this thesis techniques are developed for proving the optimality of designs with respect to the A- and (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. In particular, A-optimal designs in D(N,n) are classified for n = 6 and N arbitrary. In addition, for the cases N (TBOND) 2(mod 4) and N (TBOND) 3(mod 4), certain designs are shown to be (PHI)(,p)-optimal in D(N,n) if N is sufficiently larger than n. In the latter case, it is also shown that A-optimality of certain designs in D(N,n) implies optimality with respect to the (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. Analogous results are shown to hold for designs in D'(N,n). Further, in the case N (TBOND) 2(mod 4), designs in D'(N,n) are shown to be optimal with respect to a large class of criteria which includes the (PHI)(,p)-criteria, 0 (LESSTHEQ) p (LESSTHEQ) (INFIN), for all n and N.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1983 .M273. Source: Dissertation Abstracts International, Volume: 44-09, Section: B, page: 2778. Thesis (Ph.D.)--University of Windsor (Canada), 1983.

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