Abstract
Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs.
Similar content being viewed by others
References
Ai M, Zhang R (2004) Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285.
Bates R, Buck R, Riccomagno E, Wynn HP (1996) Experimental design and observation for large systems. J Roy Statist Soc, Ser B 58, 77–94
Fang KT, Lin DKJ, Winker P, Zhang Y (2000) Uniform design: theory and applications. Technometrics 42, 237–248
Fang KT, Ma CX, Mukerjee R (2002) Uniformity in fractional factorials. In: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, eds by Fang KT, Hickernell FJ Niederreiter H, Springer-Verlag, Berlin
Fang KT, Mukerjee R (2000) A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87, 173–198
Fang KT, Wang Y (1994) Number-Theoretic Methods in Statistics. Chapman and Hall, London
Franklin MF (1984) Constructing tables of minimum aberration p n−m designs. Technometrics 26, 225–232
Fries A, Hunter WG (1980) Minimum aberration 2k−p designs. Technometrics 22, 601–608
Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comp 67, 299–322
Hickernell FJ (1998b) Lattice Rules: How Well Do They Measure Up? In: Random and Quasi-Random Point Sets, eds by Hellekalek P, Larcher G, Lecture Notes in Statistics, Vol. 138, 109–166. Springer, New York
Hickernell FJ (2000) What affects the accuracy of quasi-Monte Carlo quadrature? In: Monte Carlo and Quasi-Monte Carlo Methods 1998, eds by Niederreiter H, Spanier J, 16–55. Springer-Verlag, Berlin
Hickernell FJ, Liu MQ (2002) Uniform designs limit aliasing. Biometrika 89, 893–904
Ma CX, Fang KT (2001) A note on generalized aberration factorial designs. Metrika 53, 85–93
MacWilliams FJ, Sloane NJA (1977) The theory of error-correcting codes. North-Holland, Amsterdam
Qin H, Fang KT (2004) Discrte discrepancy in factorial designs. Metrika 60, 59–72
Sloane NJA, Stufken J (1996) A linear programming bound for orthogonal arrays with mixed levels. J Statist Plann Infer 56, 295–305
Tang B (2001) Theory of J-characteristics for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika 88, 401–407
Tang B, Deng LY (1999) Minimum G2-aberration for nonregular fractional designs. Ann Statist 27, 1914–1926
Wang Y, Fang KT (1981) A note on uniform distribution and experimental design. Chin Sci Bull 26, 485–489
Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Statist 29, 549–560
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the NNSF of China (No. 10441001), the Key Project of Chinese Ministry of Education (No. 105119) and the Project-sponsored by SRF for ROCS (SEM) (No. [2004]176).
Rights and permissions
About this article
Cite this article
Qin, H., Ai, M. A note on the connection between uniformity and generalized minimum aberration. Statistical Papers 48, 491–502 (2007). https://doi.org/10.1007/s00362-006-0350-7
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00362-006-0350-7