# What are the merits of pseudo random numbers over quasi random numbers in monte-carlo simulation?

I understand that quasi-random numbers have much better convergence, but are there any reasons for me to use pseudo-random numbers instead?

• could you provide a link for "quasi-random numbers have much better convergence" ? – Probilitator Apr 1 '14 at 9:49
• Hi, here's the link. It converges faster because of the use of low discrepancy series. – Twilight Sparkle Apr 1 '14 at 10:02
• This should probably be on Cross Validated. – Raphael Apr 1 '14 at 14:28
• @Raphael Why should it go to Cross Validated? – Probilitator Apr 3 '14 at 8:34

Quasi Random Numbers are more tricky than it might seem, using them as a black box like with PRNGs is risky. E.g. an unscrambled Sobol' sequence is uniform only asymptotically, while for realistic sample sizes there are subvolumes with significantly different densities. You often do not realize that because the convergence graph looks good anyway, it gives no clue of the bias, and worse the low dimensional marginals $are$ indeed uniform, thus masking the problem. The same holds for lattice rules and other sequences, where empty spaces might interact with important features of the integrand.

On the other hand a good PRNG gives an unbiased result by default (see also here).

A useful compromise is to use randomized QMC, that is multiple QMC point sets shifted by a random offset. That way you get an unbiased estimator and also an error estimate; the downside is of course that convergence will not be as fast as for QMC.

Only use QMC if you know well what you're doing.

• Thank you very much, Quartz. For anyone interested in reading more, I've found this – Twilight Sparkle Apr 1 '14 at 15:13
• @Quartz - you are the boss ;) – Probilitator Apr 1 '14 at 17:57

I would argue (this is also what Quartz already hinted at) that PRNGs are far easier to set up than a well functioning QMC and are thus generally user-friendlier

Excel and R both offer a PRNG. (but not a QMC) Thus someone working with these software will be more likely to use a PRNG than to painstakingly implement a QMC. Also as Quartz explained one needs a certain level of Know-How to understand the intricacies of Quasi Monte Carlo. Not many people have that or a willing to invest the time and effort if a PRNG-approach also works (albeit not so fast)

Giuseppe Bruno, Bank of Italy, did some interesting work in R showing that the use of Quasi Random Numbers in Monte Carlo simulations was superior to Pseudo-Random. Here is an abstract of what he presented at useR! 2014: Pricing Credit Risk Derivative with R