402 THE HANDBOOK OF PORTFOLIO MATHEMATICS
required for a givensandx. Therefore, if we simply setpto .5, we are being
conservative, and requiring that (12.10) err on the side of conservatism (i.e.,
as a larger sample size).
Simply put then, we need only answer forsandx. So if I want to find
the sample size that would give me an error of .001, with a confidence tos
standard deviations, solving for (12.10) yields the following:
2 sigma=(
2
. 001
) 2
.^5 (^1 −.^5 )=1,000,000
3 sigma=(
3
. 001
) 2
.^5 (^1 −.^5 )=2,250,000
5 sigma=(
5
. 001
) 2
.^5 (^1 −.^5 )=6,250,000
Now the reader is likely to inquire, “Are these sample sizes independent
of the actual population size?” The sample sizes for the given parameters
to (12.10) will be the same regardless of whether we are trying to estimate
a population of 1,000 or 10,000,000.
“So I need only do this once; I don’t need to keep increasingq?”
Not so. Rather, you use (12.10) to discern the minimum sample size re-
quired at eachq. You still need to subsequently increaseq, and the answer
[as provided by (12.05), (12.05a), or (12.05b)] will keep increasing to the
asymptote. The reason you must keep increasingqis that at eachq, the bi-
nomial distribution is different, as was demonstrated earlier in this chapter.
One of the key caveats in implementing Equation (12.10) is that it is pro-
vided for a “random” sample size. However, these minimum, random sample
sizes provided for in (12.10) tend to be rather large. Thus, it’s important to
make sure, since we are generating random numbers by computer, that we
are not cycling in our random numbers so soon that it will cause distortion
in randomness, and that the random numbers generated be isotropically
distributed.
I strongly suggest to the ambitious readers who attempt to program
these concepts that they incorporate the most powerful random number
generators they can. Over the years this has been something of a moving
target, and, likely and hopefully will continue to be. Currently, I am partial to
the Mersenne Twister algorithm.^7 You can use other random number gener-
ators, but your results will be accurate only to the extent of the randomness
provided by them.
(^7) Makato Matsumoto and Takuji Nishimura, “Mersenne Twister: A 623-Dimensionally
Equidistributed Uniform Pseudo-Random Number Generator,”ACM Transactions
on Modeling and Computer Simulation,Vol. 8, No. 1 (January 1998), pp. 3–30.