Exact A-optimal first-order saturated designs with n ≡ 3 mod 4 observations

The results of Sathe and Shenoy (1989) on A-optimality are extended for the cases where the given lower bound is not attained. It is proved that for the saturated designs with n = 11 and n = 15 observations, this lower bound is not attained. The saturated designs R̂n with n = 11 and n = 15 observations are given and proved to be the A-optimal designs. © 1994

Extension and necessity of Cheng and Wu conditions

Repeated Measurement Designs, with two treatments, n (experimental) units and p periods are examined. The model examined is with uncorrelated observations following a continuous distribution with constant variance and the parameters of interest are (i) the difference of direct effects and (ii) the difference of residual effects. In this paper (a) the difference of Universal optimality and Φ-optimality is clarified and (b) the sufficient conditions of Cheng and Wu (1980) are extended to include the case n=2. mod. 4, p even, (c) also it is shown that these conditions are also necessary for Φ-optimality for estimating direct as well as residual effects, and (d) a method is proposed to construct Φ-optimal designs and examples are given when n even and p=3, n=0. mod. 4 and p=4, n=2. mod. 4 and p=4. In the last case the estimated parameters in the optimal design are correlated. © 2012 Elsevier B.V

Construction of some Hadamard matrices with maximum excess

AbstractLet σ(n) be the maximum excess of an Hadamard matrix of order n. Hadamard matrices with maximum excess are constructed for the following cases: (i) n = (4m + 1)2 + 3 for m = 8, 13, 18 with σ(n) = n(4m + 1) = nn − 3, (ii) n = (4m + 3)2 + 3 for m = 3, 4, 5, 12 with σ(n) = n(4m + 3) = nn − 3, (iii) n = 4(2m + 1)2 for m = 10, 16, 18 with σ(n) = n(4m + 2) = nn

CONSTRUCTION OF SOME HADAMARD-MATRICES WITH MAXIMUM EXCESS

Let sigma(n) be the maximum excess of an Hadamard matrix of order n. Hadamard matrices with maximum excess are constructed for the following cases: (i) n = (4m+1)2+3 for m = 8, 13, 18 with sigma(n) = n(4m+1) = n square-root n-3, (ii) n = (4m+3)2+3 for m = 3, 4, 5, 12 with sigma(n) = n(4m+3) = n square-root n-3, (iii) n = 4(2m+1)2 for m = 10, 16, 18 with sigma(n) = n(4m+2) = n square-root n

Estimability of parameters in a linear model and related characterizations

An alternative criterion is presented for a linear function of the unknown parameters, in a linear model, to be estimable. If there is a linear relationship between members of a subset of the columns of the design matrix X, then the parameters associated with all the columns in that subset are not estimable. Also any part of their linear relationship is non-estimable. © 2008 Elsevier B.V. All rights reserved

The concept of majorization in experimental design

Row and column designs are examined and the concept of majorization is applied to find optimal designs, when the observations are either independent or dependent. The dependence follows a first-order autoregression with parameter α. The case of two treatments is examined and the universally optimal or φ-optimal designs are given, for different values of a, when the number of experimental units is even or odd. A filtering procedure is followed to reduce the number of competing designs. Copyright © Taylor & Francis Group, LLC