Page 222 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes However, more can be done in this context than is often thought. Again, we must remember that
we are only interested in making a rough estimate. Thus, we should be ready to make back-of-
the-envelope calculations.
To illustrate what can be done, suppose that we focus on the the relative-width criterion. With
the relative-width criterion, it suffices to estimate the squared coefficient of variation (SCV,
variance divided by the square of the mean)

 2
2
c 
 2
instead of both  and  . With the relative-width criterion, the required sample size is:
2
2 2
4c z


n ( , )   /2
r 2

From the analysis above, we see that we only need to estimate a single parameter, the SCV c , in
2
order to carry out this preliminary analysis. In many cases, we can make reasonable estimates
based on “engineering judgment”. For that step, it helps to have experience with variability as
quantified by the SCV. First, note that the SCV measures the level of variability independent of
the mean: The SCV of a random variable is unchanged if we multiply the random variable by a
constant. We are thus focusing on the variability independent of the mean. Clearly, it is important
to realize that the mean itself plays no role with the relative- width criterion.
Once we learn to focus on the SCV, we quickly gain experience about what to expect. A common
reference case for the SCV is an exponential distribution, which has c = 1. A unit point mass
2
2
(deterministic distribution) has c = 0. Distributions relatively more (less) variable than
2
exponential have c > (<)1. In many instances we actually have a rough idea about the SCV. We
might be able to judge in advance that the SCV is one of: (i) less than 1, but probably not less than
0.1, (ii) near 1, (iii) bigger than 1, but probably not bigger than 10, or (iv) likely to be large, even
bigger than 10. In other words, it is not unreasonable to be able to estimate the SCV to within an
order of magnitude (within a factor of 10). And that may be good enough for the desired rough
estimates we want to make.

In lieu of information about the SCV, to obtain a rough estimate we can just let c = 1. To proceed,
2
we can also let  = 0.05, so that z a /2  2 . Then, if we set  = 10 , the required simulation run
—k
length is
–k
2k
n (10 , 0.05) = 16  10 :
r
2
Thus, when c = 1, 10% relative precision requires about 1600 replications, while 1% relative
precision requires 160,000 replications. If c  1, then we would multiply the number of required
2
replications above by c . We thus can easily factor in the only unknown, the SCV c .
2
2
We have just reviewed the classical IID case, which is directly relevant when we use
independent replications. However, in this chapter we concentrate on the more complicated
case in which we consider an initial segment of a stochastic process from a single simulation run.
It is good to have the classical IID case in mind, though, to understand the impact of bias and
dependence upon the required computational effort.
An Initial Segment from a Single Run

Now suppose that we intend to estimate a long-run average within a single run by the sample
mean from an initial segment of a stochastic process, which could evolve in either discrete time
or continuous time. The situation is now much more complicated, because the random

216 LOVELY PROFESSIONAL UNIVERSITY