Performs a d 2 test.

Routine DSQAR computes the d2 test for succeeding quadruples of hypothesized pseudorandom uniform (0, 1) deviates. The d2 test is performed as follows. Let X1, X2, X3, and X4 denote four pseudorandom uniform deviates, and consider

The probability distribution of D2 is given as

when D2 ≤ 1, where π denotes the value of pi. If D2 > 1, this probability is given as

See Gruenberger and Mark (1951) for a derivation of this distribution.

For each succeeding set of 4 pseudorandom uniform numbers input in X, d2 and the cumulative probability of d 2 (Pr(D2 ≤ d 2)) are computed. The resulting probability is tallied into one of k = NCELL equally spaced intervals.

Let n denote the number of sets of four random numbers input (n = the total number of observations/4). Then, under the null hypothesis that the numbers input are random uniform (0, 1) numbers, the expected value for each element in COUNT is e = n/k. An approximate chi‑squared statistic is computed as

where oi = COUNT(i) is the observed count. Thus, X2 has k ‑ 1 degrees of freedom, and the null hypothesis of pseudorandom uniform (0, 1) deviates is rejected if X2 is too large. As n increases, the chi‑squared approximation becomes better. A useful generalization is that e > 5 yields a good chi‑squared approximation.

Informational errors

In the following example, 2000 observations generated via routine RNUN (see Chapter 18, “Random Number Generation”) are input to DSQAR in one call. In the example, the null hypothesis of a uniform distribution is not rejected.