Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Due to the wide applications of normal random variates and the fact that applications are usually affected by their accuracy, an extensive study of the quality evaluation of unit uniform random number generators and unit normal transformation algorithms that are applications specific is recommended. In most simulation applications it is implicitly assumed that if any "good" unit uniform random number generator is used in combination with any "good" transformation algorithm. The resulting random variates will have good properties of randomness and will have the desired fit. The results of this research shows that this assumption is false. Use of good, theoretically exact transformation in conjunction with a theoretically good unit uniform generator does not necessary guarantee that the resulting variates have good statistical properties. In addition, a generator is good for one type of transformations, but it is not necessarily good for the other type of transformations. Hence, the simulation practitioner must be cautious in the selection of both the unit uniform generator and the transformation algorithm to be used. This study is concerned with the empirical study of quality evaluations of several different combinations of unit uniform random number generators and unit normal transformation algorithms for practical applications, and with procedures for determining if a given generator or a given combination is "good" or "bad". Major emphasis of this study is placed upon selection of the appropriate statistical tests for this evaluation, and development of the pertinent procedures as well as the corresponding computer programs for these tests. The statistical tests selected as appropriate for evaluating and comparing the quality of various generators are: chi-square goodness of fit tests for testing the desired distribution fits and other goodness of fits procedures for use in conjunction with other tests, runs up and down, runs above and below the median, autocorrelation tests, and spectral tests for randomness and independence. The corresponding procedures and the computer programs are set up for each test. Seven widely used unit uniform random number generators (ADRAND, RANDU, L & L, M & M, R. Shore, URAND and GGUBS) and three well known normal transformation algorithms (Box-Muller partial inverse, Box-Muller rejection, and Hasting inverse routine) are investigated here. For each generator and each combination, a large sample size series is generated which contains 50 independent samples with 50 different initial seed numbers. Each sample contains 50 sets, and each set contains 1200 generated numbers. Each selected generator and each selected combination is assessed by statistically testing the quality, both locally and globally, of the corresponding generated sequence. In total, we empirically evaluate (at significance levels 0.01, 0.05 and 0.2 respectively), the quality of seven generators, twenty-one combinations, and two IBM IMSL library unit normal generators: GGNML and GGNPM. In addition, the statistics selected and the procedures developed as well as the corresponding computer programs established here are not restricted to use in this study. They can also be applied in assessing other generators and other combinations.