Birthday Spacings Test

This test first appears in DIEHARD test suite by Marsaglia [14]. We did not implement this particular test, however we formatted the output of our RNGs to conform to the format expected by the DIEHARD suite (this was quite an exercise in itself). Out of curiosity, we then ran the Birthday Spacings Test on our RNGs to see how they performed. We thought this test was interesting, and have included the results in this report.

First choose m birthdays in a ``year'' of n days. Then, list the spacings between the m birthdays. Let j be the number of values that occur more than once in that list, then j is asymptotically Poisson distributed with mean $m^\frac{3}{4n}$. Experience shows n must be quite large, say $n \ge 2^{18}$, for comparing the results to the Poisson distribution with that mean. This test uses $n=2^{24}$ and $m=2^{10}$, so that the underlying distribution for j is taken to be Poisson with $\lambda=\frac{2^{30}}{2^{26}}=16$. A sample of 200 j''s is taken, and a $\chi ^2$ test provides a p value. The first test uses bits 1-24 (counting from the left) from integers in the specified file. Then the file is closed and reopened, then bits 2-25 of the same integers are used to provide birthdays, and so on to bits 9-32. Each set of bits provides a p-value, and the nine p-values provide a sample for a KS Test. In our analysis we examine only the $\chi ^2$ test values.