A good RNG ought to pass several statistical tests of randomness. For example, uniform real values in the range 0 to 1 can be binned into a histogram with roughly equal counts in each bin, give or take some due to statistical fluctuations. These counts obey some distribution, I don’t recall offhand if it’s Poisson or binomial or what, but in any case these distributions have tails. Same idea applies to tests for correlations, subtle periodicities etc.
A high quality RNG will occasionally fail a statistical test. It is good advice to be suspicious of RNGs that look to perfect.
Well, I’m crazy and would like to generate (reproducibly) “too perfect” random numbers, ones suspiciously lacking in those random fluctuations in statistical measures. Histograms come out too flat, variances of moving-box averages come out too small, correlations suspiciously close to zero, etc. Looking for RNGs that pass all statistical tests too cleanly. What known RNGs are like this? Is there published research on this idea?
One unacceptable answer: some of the poorer linear congruential counter generators have too flat a distribution, but totally flunk most tests of randomness.
Related to this is the generation of random number streams with a known calibrated amount of imperfection. A lump in the distribution is easy – just generate a nonuniform distribution approximating the idea (e.g see Generating non-uniform random numbers) but what about introducing calibrated amounts of higher order correlations while maintaining a correct, or too perfect, distribution?
Apparently the Mersenne Twister, a commonly used random number generator, fails the DieHarder tests by being “too random”. In other words, certain tests consistently come too close to their expected value under true randomness.