Subject the canned random-number generator on your computer to the chi-square test, two-and three-dimensional serial tests, the runs-up test, and correlation tests at lags 1, 2, ..., 5. Use the same values for n, k, and a that were used in Examples 7.6 through 7.9. (If your generator does not pass these tests, we suggest that you exercise caution in using it until you can obtain more information on it, either from the literature or from your own further testing.) Example 7.6 EXAMPLE 7.6. We applied the chi-square test of uniformity to the PMMLCG Z 630,360,0167,-,(mod 2 - 1), as implemented in App. 7A, using stream 1 with the default seed. We took k22 - 4096 (so that the most significant 12 bits of the U's are being examined for uniformity) and let n = 215 - 32,768. We obtained x-4141.0, using the above approximation for the critical point, Xos.0.5 4245.0, so the null hypothesis of uniformity is not rejected at level a = 0.05. Therefore, these particular 32,768 U,'s produced by this generator do not behave in a way that is significantly dif- ferent from what would be expected from truly IID U(0, 1) random variables, so far as this chi-square test can ascertain. Example 7.7 EXAMPLE 7.7. For d - 2, we tested the null hypothesis that the pairs (U, U), (U, U.)....(UM-,U) are IID random vectors distributed uniformly over the unit square. We used the generator in App. 7A, but starting with stream 2, and generated = 32.768 pairs of Us. We took k = 64, so that the degrees of freedom were again 4095 642 - 1 and the level a 0.05 critical value was the same, 4245.0. The value of X(2) was 4016.5. indicating acceptable uniformity in two dimensions for the first Iwo-thirds of stream 2 (recall from Sec. 2.3 that the streams are of length 100,000 US. and we used 2n = 65.536 of them here). For d = 3. we used stream 3. took k = 16 (keeping the degrees of freedom as 4095 - 16 - 1 and the level a = 0.05 critical value at 4245.0), and generated n = 32,768 nonoverlapping triples of Us. And X (3) was 4174.5. again indicating acceptable uniformity in three dimensions. Example 7.8 EXAMPLE 7.8. We subjected stream 4 of the generator in App. 7A to the runs-up test, using n = 5000, and obtained (rig...T) = (808, 1026, 448, 139, 43, 4), leading to a value of R = 9.3. Since xo s = 12.6, we do not reject the hypothesis of indepen- dence at level a = 0.05. Example 7.9 EXAMPLE 7.9. We tested streams 5 through 10 of the generator in App. 7A for correla- tion at lags 1 through 6, respectively, taking n = 5000 in each case; i.e., we tested stream 5 for lag 1 correlation, stream 6 for lag 2 correlation, etc. The values of A, A2,...A were 0.90, -1.03-0.12. 1.32, 0.39, and 0.76, respectively, none of which is significantly different from 0 in comparison with the No. 1) distribution, at level a = 0.05. Thus, the first 5000 values in these streams do not exhibit observable autocorrelation at these lags Subject the canned random-number generator on your computer to the chi-square test, two-and three-dimensional serial tests, the runs-up test, and correlation tests at lags 1, 2, ..., 5. Use the same values for n, k, and a that were used in Examples 7.6 through 7.9. (If your generator does not pass these tests, we suggest that you exercise caution in using it until you can obtain more information on it, either from the literature or from your own further testing.) Example 7.6 EXAMPLE 7.6. We applied the chi-square test of uniformity to the PMMLCG Z 630,360,0167,-,(mod 2 - 1), as implemented in App. 7A, using stream 1 with the default seed. We took k22 - 4096 (so that the most significant 12 bits of the U's are being examined for uniformity) and let n = 215 - 32,768. We obtained x-4141.0, using the above approximation for the critical point, Xos.0.5 4245.0, so the null hypothesis of uniformity is not rejected at level a = 0.05. Therefore, these particular 32,768 U,'s produced by this generator do not behave in a way that is significantly dif- ferent from what would be expected from truly IID U(0, 1) random variables, so far as this chi-square test can ascertain. Example 7.7 EXAMPLE 7.7. For d - 2, we tested the null hypothesis that the pairs (U, U), (U, U.)....(UM-,U) are IID random vectors distributed uniformly over the unit square. We used the generator in App. 7A, but starting with stream 2, and generated = 32.768 pairs of Us. We took k = 64, so that the degrees of freedom were again 4095 642 - 1 and the level a 0.05 critical value was the same, 4245.0. The value of X(2) was 4016.5. indicating acceptable uniformity in two dimensions for the first Iwo-thirds of stream 2 (recall from Sec. 2.3 that the streams are of length 100,000 US. and we used 2n = 65.536 of them here). For d = 3. we used stream 3. took k = 16 (keeping the degrees of freedom as 4095 - 16 - 1 and the level a = 0.05 critical value at 4245.0), and generated n = 32,768 nonoverlapping triples of Us. And X (3) was 4174.5. again indicating acceptable uniformity in three dimensions. Example 7.8 EXAMPLE 7.8. We subjected stream 4 of the generator in App. 7A to the runs-up test, using n = 5000, and obtained (rig...T) = (808, 1026, 448, 139, 43, 4), leading to a value of R = 9.3. Since xo s = 12.6, we do not reject the hypothesis of indepen- dence at level a = 0.05. Example 7.9 EXAMPLE 7.9. We tested streams 5 through 10 of the generator in App. 7A for correla- tion at lags 1 through 6, respectively, taking n = 5000 in each case; i.e., we tested stream 5 for lag 1 correlation, stream 6 for lag 2 correlation, etc. The values of A, A2,...A were 0.90, -1.03-0.12. 1.32, 0.39, and 0.76, respectively, none of which is significantly different from 0 in comparison with the No. 1) distribution, at level a = 0.05. Thus, the first 5000 values in these streams do not exhibit observable autocorrelation at these lags