The Annals of Statistics

Optimum and Minimax Exact Treatment Designs for One-Dimensional Autoregressive Error Processes

J. Kiefer and H. P. Wynn

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A theory is developed following work by Williams (1952) and Kiefer (1960) for exact treatment designs in one dimension in which the errors are a stationary process. It is shown that the designs which achieve the minimax value of any of a wide class of functionals on the information matrix for estimation of treatment differences have a special property. If the process is autoregressive of order $p$ then a random piece of the design of length $p + 1$ exhibits uncorrelated treatment values. Such designs can be formed using full length cyclic error-correcting codes of a suitable order. A new technique is developed for classifying the ergodic combinatorial structure of exact designs of arbitrary or infinite length. It is shown that all designs are, to $p$th order, generated by a finite number of sequences with finite length. The classification is given explicitly up to order 3. The method is used to find asymptotically optimum designs for different processes. It is also shown that the designs can be achieved to within an arbitrarily good approximation as the realization of an ergodic Markov chain of sufficiently high order.

Article information

Ann. Statist., Volume 12, Number 2 (1984), 431-450.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62K05: Optimal designs
Secondary: 62K15: Factorial designs 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 05B15: Orthogonal arrays, Latin squares, Room squares 62J05: Linear regression

Optimum experimental designs exact design dependent observations stationary processes binary sequences error correcting codes linear machines linear programming pseudo-random sequences Markov chains


Kiefer, J.; Wynn, H. P. Optimum and Minimax Exact Treatment Designs for One-Dimensional Autoregressive Error Processes. Ann. Statist. 12 (1984), no. 2, 431--450. doi:10.1214/aos/1176346498.

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