## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 29, Number 3 (1958), 675-699.

### On the Nonrandomized Optimality and Randomized Nonoptimality of Symmetrical Designs

#### Abstract

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.

#### Article information

**Source**

Ann. Math. Statist. Volume 29, Number 3 (1958), 675-699.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoms/1177706530

**Digital Object Identifier**

doi:10.1214/aoms/1177706530

**Mathematical Reviews number (MathSciNet)**

MR98451

**Zentralblatt MATH identifier**

0092.36102

**JSTOR**

links.jstor.org

#### Citation

Kiefer, J. On the Nonrandomized Optimality and Randomized Nonoptimality of Symmetrical Designs. Ann. Math. Statist. 29 (1958), no. 3, 675--699. doi:10.1214/aoms/1177706530. http://projecteuclid.org/euclid.aoms/1177706530.