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Centered $L_2$-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs

Authors: Kai-Tai Fang, Chang-Xing Ma and Peter Winker
Journal: Math. Comp. 71 (2002), 275-296
MSC (2000): Primary 68U07; Secondary 65D17, 62K99
Published electronically: October 16, 2000
MathSciNet review: 1863000
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Abstract: In this paper properties and construction of designs under a centered version of the $L_2$-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

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Additional Information

Kai-Tai Fang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong; and Chinese Academy of Sciences, Beijing, China

Chang-Xing Ma
Affiliation: Department of Statistics, Nankai University, Tianjin, China

Peter Winker
Affiliation: Department of Economics, University of Mannheim, 68131 Mannheim, Germany

Keywords: Uniform design, Latin hypercube design, threshold accepting heuristic, quasi-Monte Carlo methods
Received by editor(s): July 20, 1999
Received by editor(s) in revised form: February 25, 2000
Published electronically: October 16, 2000
Additional Notes: This work was partially supported by a Hong Kong RGC-grant and SRCC of Hong Kong Baptist University.
Article copyright: © Copyright 2000 American Mathematical Society