## The Annals of Statistics

- Ann. Statist.
- Volume 9, Number 4 (1981), 737-757.

### Optimum Balanced Block and Latin Square Designs for Correlated Observations

J. Kiefer and H. P. Wynn

#### Abstract

In this paper designs are found which are optimum for various models that include some autocorrelation in the covariance structure $V$. First it is noted that the ordinary least squares estimator is quite robust against small perturbations in $V$ from the uncorrelated case $V_0 = \sigma^2_I$. This "local" argument justifies our use of such estimators and restriction to the class of designs $\mathscr{X}^\ast$ (balanced incomplete block or Latin squares) optimum under $V_0$. Within $\mathscr{X}^\ast$ we search for designs for which the least squares estimator minimizes appropriate functionals of the dispersion matrix under various correlation models $V$. In particular, we consider "nearest neighbor" correlation models in detail. The solutions lead to interesting combinatorial conditions somewhat similar to those encountered in "repeated measurement" designs. Typically, however, the latter need not be BIBD's and require twice as many blocks. For Latin squares, and hypercubes, the conditions are less restrictive than those giving "completeness."

#### Article information

**Source**

Ann. Statist., Volume 9, Number 4 (1981), 737-757.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345515

**Digital Object Identifier**

doi:10.1214/aos/1176345515

**Mathematical Reviews number (MathSciNet)**

MR624701

**Zentralblatt MATH identifier**

0546.62051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62K05: Optimal designs

Secondary: 05B20: Matrices (incidence, Hadamard, etc.)

**Keywords**

Optimum designs correlated observations difference sets balanced incomplete block designs Latin squares and hypercubes

#### Citation

Kiefer, J.; Wynn, H. P. Optimum Balanced Block and Latin Square Designs for Correlated Observations. Ann. Statist. 9 (1981), no. 4, 737--757. doi:10.1214/aos/1176345515. https://projecteuclid.org/euclid.aos/1176345515