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September, 1958 On the Nonrandomized Optimality and Randomized Nonoptimality of Symmetrical Designs
J. Kiefer
Ann. Math. Statist. 29(3): 675-699 (September, 1958). DOI: 10.1214/aoms/1177706530


Many commonly employed symmetrical designs such as Balanced Incomplete Block Designs (BIBD's), Latin Squares (LS's), Youden Squares (YS's), etc., are shown to have optimum properties among the class of non-randomized designs (Section 3). This represents an extension of a property first proved by Wald for LS's in [1]; a similar property demonstrated by Ehrenfeld for LS's in [2] (as well as a third optimum property considered here) is shown to be an immediate consequence of the Wald property, and the Wald property is shown to be the more relevant when one considers optimality rigorously (Section 2). Surprisingly, all of these optimum properties fail to hold if randomized designs are considered (Section 4); the results of Sections 2 and 3, as well as those appearing previously in the literature (as in [1], [2], [3]) must be interpreted in this sense. Generalizations of the BIBD's and YS's, for which analogous results hold, are introduced.


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J. Kiefer. "On the Nonrandomized Optimality and Randomized Nonoptimality of Symmetrical Designs." Ann. Math. Statist. 29 (3) 675 - 699, September, 1958.


Published: September, 1958
First available in Project Euclid: 27 April 2007

zbMATH: 0092.36102
MathSciNet: MR98451
Digital Object Identifier: 10.1214/aoms/1177706530

Rights: Copyright © 1958 Institute of Mathematical Statistics

Vol.29 • No. 3 • September, 1958
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