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Regular Fractions of Factorial Arrays

  • Ulrike GrömpingEmail author
  • R. A. Bailey
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

For symmetric arrays of two-level factors, a regular fraction is a well-defined concept, which has been generalized in various ways to arrays of s-level factors with s a prime or prime power, and also to mixed-level arrays with arbitrary numbers of factor levels. This paper introduces three further related definitions of a regular fraction for a general array, based on squared canonical correlations or the commuting of projectors. All classical regularity definitions imply regularity under the new definitions, which also permit further arrays to be considered regular. As a particularly natural example, non-cyclic Latin squares, which are not regular under several classical regularity definitions, are regular fractions under the proposed definitions. This and further examples illustrate the different regularity concepts.

Notes

Acknowledgements

Ulrike Grömping’s initial work was supported by Deutsche Forschungsgemeinschaft (Grant GR 3843/1-1). Ulrike Grömping wishes to thank Hongquan Xu for fruitful discussions on an earlier version of this work. The collaboration with Rosemary Bailey was initiated at a workshop funded by Collaborative Research Center 823 at TU Dortmund University.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Beuth University of Applied Sciences BerlinBerlinGermany
  2. 2.School of Mathematics & StatisticsUniversity of St AndrewsSt AndrewsUK

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