Further details on majorization can be found in Marshall et al. If x is majorized by every competing y, then the design producing x is said to be M-optimal among all competing designs. The relationship between x and y expressed in ( A.1) is termed “ x is weakly majorized from above by y,” which will often be shortened to “ x is majorized by y.” In the applications here, x and y are vectors of \(v-1\) nonzero eigenvalues arising from competing designs, and ( A.1) says the design producing x is M-better than that producing y. Here \(x^\uparrow \) denotes the vector obtained by rearranging the co-ordinates of x in non-decreasing order. Tomić M (1949) Théorème de Gauss relatif au centre de gravité et son application. Shah KR, Sinha BK (1989) Theory of Optimal Designs. Russell KG (1980) Further results on the connectedness and optimality of designs of type \(O:XB\). Oehlert GW (2000) A First Course in Design and Analysis of Experiments. Morgan JP, Stallings JW (2014) On the \(A\) criterion of experimental design. ![]() Morgan JP, Srivastav SK (2000) On the type-1 optimality of nearly balanced incomplete block designs with small concurrence range. Morgan JP, Reck B (2007) E-optimal design in irregular BIBD settings. Morgan JP, Jermjitpornchai S (2021) Optimal row-column designs with three rows. Handbook of Design and Analysis of Experiments. ![]() Morgan JP (2015) Blocking with independent responses. Springer Series in Statistics, Springer, New York Marshall AW, Olkin I, Arnold BC (2011) Inequalities: Theory of Majorization and its Applications, 2nd edn. Kiefer J (1975) Construction and optimality of generalized Youden designs. Jacroux M (1985) Some sufficient conditions for the type \(1\) optimality of block designs. Jacroux M (1982) Some E-optimal designs for the one-way and two-way elimination of heterogeneity. ![]() Ann Statist 6(6):1239–1261ĭean A, Voss D, Draguljić D (2017) Design Anal Experiment, 2nd edn. J Stat Theory Pract 5(1):59–67Ĭheng CS (1978) Optimality of certain asymmetrical experimental designs. Stat Probab Lett 39(2):173–177Ĭhai F-S, Cheng C-S (2011) Some optimal row-column designs. Ann Stat 29(2):577–594Ĭhai F-S (1998) A note on generalization of distinct representatives. Ann Math Stat 37:525–528īagchi B, Bagchi S (2001) Optimality of partial geometric designs. Agrawal H (1966) Some generalizations of distinct representatives with applications to statistical designs.
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