## The Annals of Statistics

- Ann. Statist.
- Volume 8, Number 2 (1980), 447-453.

### Orthogonal Arrays with Variable Numbers of Symbols

#### Abstract

Orthogonal arrays with variable numbers of symbols are shown to be universally optimal as fractional factorial designs. The orthogonality of completely regular Youden hyperrectangles ($F$-hyperrectangles) is defined as a generalization of the orthogonality of Latin squares, Latin hypercubes, and $F$-squares. A set of mutually orthogonal $F$-hyperrectangles is seen to be a special kind of orthogonal array with variable numbers of symbols. Theorems on the existence of complete sets of mutually orthogonal $F$-hyperrectangles are established which unify and generalize earlier results on Latin squares, Latin hypercubes, and $F$-squares.

#### Article information

**Source**

Ann. Statist. Volume 8, Number 2 (1980), 447-453.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176344964

**Digital Object Identifier**

doi:10.1214/aos/1176344964

**Mathematical Reviews number (MathSciNet)**

MR560740

**Zentralblatt MATH identifier**

0431.62048

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62K05: Optimal designs

Secondary: 62K15: Factorial designs 05B15: Orthogonal arrays, Latin squares, Room squares

**Keywords**

Orthogonal arrays fractional factorial designs Latin squares Latin hypercubes $F$-squares completely regular Youden hyperrectangles $F$-hyperrectangles

#### Citation

Cheng, Ching-Shui. Orthogonal Arrays with Variable Numbers of Symbols. Ann. Statist. 8 (1980), no. 2, 447--453. doi:10.1214/aos/1176344964. http://projecteuclid.org/euclid.aos/1176344964.