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Chapter 6: Monte Carlo Methods for Inferential Statistics 205

lation must be something we can sample from using the computer. In this
text, we consider this type of Monte Carlo simulation to be a parametric tech-
nique, because we sample from a known or assumed distribution.
The basic Monte Carlo procedure is outlined here. Later, we provide proce-
dures illustrating some specific uses of Monte Carlo simulation as applied to
statistical hypothesis testing.

PROCEDURE - BASIC MONTE CARLO SIMULATION

1. Determine the pseudo-population or model that represents the true
population of interest.
2. Use a sampling procedure to sample from the pseudo-population.
3. Calculate a value for the statistic of interest and store it.
4. Repeat steps 2 and 3 for M trials.
5. Use the M values found in step 4 to study the distribution of the
statistic.
It is important to keep in mind, that when sampling from the pseudo-popu-
lation, the analyst should ensure that all relevant characteristics reflect the
statistical situation. For example, the same sample size and sampling strategy
should be used when trying to understand the performance of a statistic. This
means that the distribution for the statistic obtained via Monte Carlo simula-
tion is valid only for the conditions of the sampling procedure and the
assumptions about the pseudo-population.
Note that in the last step of the Monte Carlo simulation procedure, the ana-
lyst can use the estimated distribution of the statistic to study characteristics
of interest. For example, one could use this information to estimate the skew-
ness, bias, standard deviation, kurtosis and many other characteristics.

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Recall that in statistical hypothesis testing, we have a test statistic that pro-
vides evidence that the null hypothesis should be rejected or not. Once we
observe the value of the test statistic, we decide whether or not that particular
value is consistent with the null hypothesis. To make that decision, we must
know the distribution of the statistic when the null hypothesis is true. Esti-
mating the distribution of the test statistic under the null hypothesis is one of
the goals of Monte Carlo hypothesis testing. We discuss and illustrate the
Monte Carlo method as applied to the critical value and p-value approaches
to hypothesis testing.
Recall that in the critical value approach to hypothesis testing, we are given
α
a significance level . We then use this significance level to find the appro-
priate critical region in the distribution of the test statistic when the null
hypothesis is true. Using the Monte Carlo method, we determine the critical
© 2002 by Chapman & Hall/CRC