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June, 1984 Optimum and Minimax Exact Treatment Designs for One-Dimensional Autoregressive Error Processes
J. Kiefer, H. P. Wynn
Ann. Statist. 12(2): 431-450 (June, 1984). DOI: 10.1214/aos/1176346498

Abstract

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.

Citation

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J. Kiefer. H. P. Wynn. "Optimum and Minimax Exact Treatment Designs for One-Dimensional Autoregressive Error Processes." Ann. Statist. 12 (2) 431 - 450, June, 1984. https://doi.org/10.1214/aos/1176346498

Information

Published: June, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0558.62066
MathSciNet: MR740904
Digital Object Identifier: 10.1214/aos/1176346498

Subjects:
Primary: 62K05
Secondary: 05B15 , 62J05 , 62K15 , 62M10

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

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 2 • June, 1984
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