A study on design uniformity under errors in the level values

1. Preliminary The uniform design initially proposed by Wang and Fang (1981) is a space-filling design for computer experiments, but it can also be utilized as a fractional factorial design. In the past two decades, it has been successfully applied to a wide range of areas; for example, in industry, system engineering, pharmaceuticals and natural sciences (see Fang and Lin, 2003). Many uniform designs are available at the website http://www.math.hkbu.edu.hk/UniformDesign. A U-type design U(n; q1, . . . , qs) corresponds to an n × s matrix U = (uik), such that the elements uik (i = 1, . . . , n) in the kth column take values (equally often) from a set of qk integers, say {1, . . . , qk}. When some of the qk are equal, we denote this U-type design by U(n; q1 1 · · · q sm m ), where s = ∑m k=1 sk, or U(n; q s) when all the qk are equal. By mapping uik to xik = (2uik − 1)/(2qk) for i = 1, . . . , n and k = 1, . . . , s, the n runs of a U(n; q1, . . . , qs) can be transformed to n points in C s = [0, 1]s. The uniformity of a design is usually measured by a discrepancy criterion (Hickernell, 1998a,b). There are several different discrepancies that have been defined, amongwhich thewrap-around L2-discrepancy (WD2, for short) and centered L2-discrepancy (CD2, for short) are regarded as the most practicable ones. In this paper, the wrap-around L2-discrepancy is used; it can be defined as WD2(U) = [ − (4 3 )s + 1 n2 n ∑