Abstract
In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L 2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L 2-discrepancy value of D(X) and generalized wordlength pattern of X, which show the uniformity relationship between D(X) and X. In addition, we also obtain lower bounds of centered L 2-discrepancy value of D(X), which can be used to assess uniformity of D(X).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen, H.G., Cheng, C.S. Doubling and projection: a method of constructing two-level designs of resolution IV. Ann. Statist., 34: 546–558 (2006)
Fang, K.T., Lu, X., Winker, P. Lower bounds for centered and wrap-around L 2-discrepancies and construction of uniform designs by threshold accepting. J. Complexity, 19: 692–711 (2003)
Fang, K.T., Ma, C.X., Mukerjee, R. Uniformity in fractional factorials. In: Fang, K.T., Hickernell, F.J., Niederreiter H. (eds.), Monte Carlo and Quasi-Monte Carlo Methods, Springer-Verlag, Berlin, 2002
Fang, K.T., Mukerjee, R. A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika, 87: 193–198 (2000)
Fang, K.T, Qin, H. Uniformity pattern and related criteria for two-level factorials. Science in China (Series A), 48: 1–11 (2005)
Fries, A., Hunter, W.G. Minimum aberration 2k-p designs. Technometrics, 22: 601–608 (1980)
Hickernell, F.J. A generalized discrepancy and quadrature error Bound. Mathematics of Computation, 67: 299–322 (1998)
Hickernell, F.J. Lattice Rules: How Well do They Measure Up? In: Hellekalek P, Larcher G (eds.) Random and Quasi-Random Point Sets. Lecture Notes in Statistics, Vol.138, Springer-Verlag, New York, 1998, 109–166
Lin, D.K.J. A new class of supersaturated designs. Technometrics, 35: 28–31 (1993)
Mukerjee, R, Wu, C.F.J. On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist., 23: 2102–2115 (1995)
Plackett, R.L., Burman, J.P. The design of optimum multi-factorial experiments. Biometrika, 33: 305–325 (1946)
Qin, H., Fang, K.T. Discrete discrepancy in factorial designs. Metrika, 60: 59–72 (2004)
Song, S., Qin, H. Application of minimum projection uniformity in complementary designs. Acta. Math. Sci., 30: 180–186 (2010)
Xu, H.Q., Cheng, C.S. A complementary design theory for doubling. Ann. Statist., 36: 445–457 (2006)
Xu, H.Q., Wu, C.F.J. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist., 29: 549–560 (2001)
Zhang, A.J., Fang, K.T., Li, R., Sudjianto, A. Majorization framework for balanced lattice designs. Ann. Statist., 33: 2837–2853 (2005)
Zhang, S.L., Qin, H. Minimum projection uniformity criterion and its application. Statist. Probab. Letters, 76: 634–640 (2006)
Zhou, Y.D., Ning, J.H., Song, X.B. Lee discrepancy and its applications in experimental designs. Statist. Probab. Letters, 78: 1933–1942 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (11271147) and SRFDP (20090144110002).
Rights and permissions
About this article
Cite this article
Lei, Yj., Qin, H. Uniformity in double designs. Acta Math. Appl. Sin. Engl. Ser. 30, 773–780 (2014). https://doi.org/10.1007/s10255-014-0419-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-014-0419-3