Lih-Yuan Deng

Design Selection and Classification for Hadamard Matrices Using Generalized Minimum Aberration Criteria

Deng and Tang (1999) and Tang and Deng (1999) proposed and justified two criteria of generalized minimum aberration for general two-level fractional factorial designs. The criteria are defined using a set of values called J characteristics. In this article, we examine the practical use of the criteria in design selection. Specifically, we consider the problem of classifying and ranking designs that are based on Hadamard matrices. A theoretical result on J characteristics is developed to facilitate the computation. The issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity. Our studies reveal that generalized aberration performs quite well under these familiar criteria.

Random Number Generation for the New Century

Abstract Use of empirical studies based on computer-generated random numbers has become a common practice in the development of statistical methods, particularly when the analytical study of a statistical procedure becomes intractable. The quality of any simulation study depends heavily on the quality of the random number generators. Classical uniform random number generators have some major defects—such as the (relatively) short period length and the lack of higher-dimension uniformity. Two recent uniform pseudo-random number generators (MRG and MCG) are reviewed. They are compared with the classical generator LCG. It is shown that MRG/MCG are much better random number generators than the popular LCG. Special forms of MRG/MCG are introduced and recommended as the random number generators for the new century. A step-by-step procedure for constructing such random number generators is also provided.

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators

We propose a system of multiple recursive generators of modulus <i>p</i> and order <i>k</i> where all nonzero coefficients of the recurrence are equal. The advantage of this property is that a single multiplication is needed to compute the recurrence, so the generator would run faster than the general case. For <i>p</i> = 2³¹ − 1, the most popular modulus used, we provide tables of specific parameter values yielding maximum period for recurrence of order <i>k</i> = 102 and 120. For <i>p</i> = 2³¹ − 55719 and <i>k</i> = 1511, we have found generators with a period length approximately 10^14100.5.

Period Extension and Randomness Enhancement Using High-Throughput Reseeding-Mixing PRNG

We present a new reseeding-mixing method to extend the system period length and to enhance the statistical properties of a chaos-based logistic map pseudo random number generator (PRNG). The reseeding method removes the short periods of the digitized logistic map and the mixing method extends the system period length to 2253 by “xoring” with a DX generator. When implemented in the TSMC 0.18- μm 1P6M CMOS process, the new reseeding-mixing PRNG (RM-PRNG) attains the best throughput rate of 6.4 Gb/s compared with other nonlinear PRNGs. In addition, the generated random sequences pass the NIST SP 800-22 statistical tests including ratio test and U-value test.

Efficient and portable multiple recursive generators of large order

Deng and Xu [2003] proposed a system of multiple recursive generators of prime modulus <i>p</i> and order <i>k</i>, where all nonzero coefficients of the recurrence are equal. This type of generator is efficient because only a single multiplication is required. It is common to choose <i>p</i> = 2³¹−1 and some multipliers to further improve the speed of the generator. In this case, some fast implementations are available without using explicit division or multiplication. For such a <i>p</i>, Deng and Xu [2003] provided specific parameters, yielding the maximum period for recurrence of order <i>k</i>, up to 120. One problem of extending it to a larger <i>k</i> is the difficulty of finding a complete factorization of <i>p</i>^<i>k</i>−1. In this article, we apply an efficient technique to find <i>k</i> such that it is easy to factor <i>p</i>^<i>k</i>−1, with <i>p</i> = 2³¹−1. The largest one found is <i>k</i> = 1597. To find multiple recursive generators of large order <i>k</i>, we introduce an efficient search algorithm with an early exit strategy in case of a failed search. For <i>k</i> = 1597, we constructed several efficient and portable generators with the period length approximately 10^14903.1.

Moment Aberration Projection for Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs, such as Plackett–Burman designs, are widely used in industrial experiments for run size economy and flexibility. A novel criterion, called moment aberration projection, is proposed to rank and classify nonregular designs. It measures the goodness of a design through moments of the number of coincidences between the rows of its projection designs. The new criterion is used to rank and classify designs of 16, 20, and 27 runs. Examples are given to illustrate that the ranking of designs is supported by other design criteria.

Generation of Uniform Variates from Several Nearly Uniformly Distributed Variables

A very useful result for generating random numbers is that the fractional part of a sum of independent U(0,1) random variables is also a U(0,l) random variable. In this paper we show that a more general result is true: the fractional part of a sum of n independent random variables, one of which is U(0,l), is also U(0,l). Moreover, we show that the fractional part of a sum of independent near-uniform variables is closer in distribution to a U(0,l) variate than each of the component near-uniform variables. These results are used to characterize the uniform distribution and to give some justification for an algorithm of Wichmann and Hill(1982). In addition, we show how the property of “closeness” carries over to the generation of any random variable.

Design catalog based on minimum G-aberration

Deng and Tang (Technometrics 44 (2002) 173) constructed a catalog of designs of 16, 20 and 24 runs using the criterion of minimum G-aberration, by searching through all orthogonal arrays from Hadamard matrices. Since it is not true that every orthogonal array can be embedded into a Hadamard matrix, some good designs may be missing from their catalog. This paper examines the same problem by considering all orthogonal arrays. Two advantages result from removing the restriction to designs from Hadamard matrices. We are able to obtain some results on designs of run size 28 or higher. More importantly, we have indeed found minimum G-aberration designs that cannot be embedded into Hadamard matrices. A catalog of useful designs is presented here and its usefulness discussed.

Estimation of Variance of the Regression Estimator

Abstract The regression estimator and the ratio estimator are commonly used in survey practice. In the past more attention has been given to the ratio estimator because of its computational ease and applicability for general sampling designs. The ratio estimator is appropriate for populations whose regression line passes close to the origin. If the intercept of the regression line is significantly nonzero, however, it is much less efficient than the regression estimator (Deng 1984). In general, apart from n –2 terms, the mean squared error (MSE) of the former is bigger than that of the latter (Cochran 1977, p. 196). Given the present computing capacity, the computational advantage of the ratio estimator should be less of a concern and the regression estimator will gain wider popularity. The main purpose of this article is to provide a theoretical and empirical comparison of several variance estimators for the regression estimator in simple random sampling without replacement. The companion problem for the...

Generalized Mersenne Prime Number and Its Application to Random Number Generation

A Mersenne prime number is a prime number of the form 2k — 1. In this paper, we consider a Generalized Mersenne Prime (GMP) which is of the form R(k,p) = (p k -l)/(p - 1), where k,p and R(k,p) are prime numbers. For such a GMP, we then propose a much more efficient search algorithm for a special form of Multiple Recursive Generator (MRG) with the property of an extremely large period length and a high dimension of equidistribution. In particular, we find that (p k - l)/(p - 1) is a GMP, for k = 1511 and p = 2147427929. We then find a special form of MRG with order k = 1511 and modulus p = 2147427929 with the period length 1014100.5.Many other efficient and portable generators with various k ≤ 1511 are found and listed. Finally, for such a GMP and generator, we propose a simple and quick method of generating maximum period MRGs with the same order k. The readers are advised not to confuse GMP defined in this paper with other generalizations of the Mersenne Prime. For example, the term “Generalized Mersenne Number” (GMN) is used in Appendix 6.1 of FIPS-186-2, a publication by National Institute of Standards and Technology (NIST). In that document, GMN is a prime number that can be written as 2k ± 1 plus or minus a few terms of the form 2r.