72 RANDOM NUMBERS 3.3.2 L. Historical remarks and (Free web hosting services)
72 RANDOM NUMBERS 3.3.2 L. Historical remarks and further discussion. Statistical tests arose naturally in the course of scientists efforts to prove or disprove hypotheses about various observed data. The best known early papers dealing with the testing of artificially generated numbers for randomness are two articles by M. G. Kendall and B. Babington-Smith in the Journd of the Royal Statisticd Society 101 (1938), 147-166, and in the supplement to that journal, 6 (1939), 51-61. These papers were concerned with the testing of random digits between 0 and 9, rather than random real numbers; for this purpose, the authors discussed the frequency test, serial test, gap test, and poker test, although they misapplied the serial test. Kendall and Babington-Smith also used a variant of the coupon collector s test; the method described in this section was introduced by R. E. Greenwood in Math. Comp. 9 (1955), l-4. The run test has a rather interesting history. Originally, tests were made on runs up and down at once: a run up would be followed by a run down, then another run up, and so on. Note that the run test and the permutation test do not depend on the uniform distribution of the U s, they depend only on the fact that Vi = Uj occurs with probability zero when i # j; therefore these tests can be applied to many types of random sequences. The run test in primitive form was originated by J. Bienayme [Comptes Rendus 81 (Paris: Acad. Sciences, 1875), 417-4231. S ome sixty years later, W. 0. Kermack and A. G. McKendrick published two extensive papers on the subject (Proc. Royal Society Edinburgh 57 (1937), 228-240, 332-3761; as an example they stated that Edinburgh rainfall between the years 1785 and 1930 was entirely random in character with respect to the run test (although they examined only the mean and standard deviation of the run lengths). Several other people began using the test, but it was not until 1944 that the use of the chi-square method in connection with this test was shown to be incorrect. The paper by H. Levene and J. Wolfowitz in Annals Math. Stat. 15 (1944), 58-69, introduced the correct run test (for runs up and down, alternately) and discussed the fallacies in earlier misuses of that test. Separate tests for runs up and runs down, as proposed in the text above, are more suited to computer application, so we have not given the more complex formulas for the alternate-up-and-down case. See the survey paper by D. E. Barton and C. L. Mallows, Annals Math. Stat. 36 (1965), 236-260. Of all the tests we have discussed, the frequency test and the serial correla- tion test seem to be the weakest, in the sense that nearly all random number generators pass these tests. Theoretical grounds for the weakness of these tests are discussed briefly in Section 3.5 (cf. exercise 3.5-26). The run test, on the other hand, is a rather strong test: the results of exercises 3.3.3-23 and 24 sug- gest that linear congruential generators tend to have runs somewhat longer than normal if the multiplier is not large enough, so the run test of exercise 14 is definitely to be recommended. The collision test is also highly recommended, since it has been especially designed to detect the deficiencies of many poor generators that have unfor- tunately become widespread. This test, which is based on ideas of H. Delgas Christiansen [Inst. Math. Stat. and Oper. Res., Tech. Univ. Denmark (Oct. 1975),
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