Page 204 - Computational Statistics Handbook with MATLAB

Chapter 6

Monte Carlo Methods for Inferential Statistics

6.1 Introduction
Methods in inferential statistics are used to draw conclusions about a popu-
lation and to measure the reliability of these conclusions using information
obtained from a random sample. Inferential statistics involves techniques
such as estimating population parameters using point estimates, calculating
confidence interval estimates for parameters, hypothesis testing, and model-
ing (e.g., regression and density estimation). To measure the reliability of the
inferences that are made, the statistician must understand the distribution of
any statistics that are used in the analysis. In situations where we use a well-
understood statistic, such as the sample mean, this is easily done analytically.
However, in many applications, we do not want to be limited to using such
simple statistics or to making simplifying assumptions. The goal of this chap-
ter is to explain how simulation or Monte Carlo methods can be used to make
inferences when the traditional or analytical statistical methods fail.
According to Murdoch [2000], the term Monte Carlo originally referred to
simulations that involved random walks and was first used by Jon von Neu-
mann and S. M. Ulam in the 1940’s. Today, the Monte Carlo method refers to
any simulation that involves the use of random numbers. In the following
sections, we show that Monte Carlo simulations (or experiments) are an easy
and inexpensive way to understand the phenomena of interest [Gentle, 1998].
To conduct a simulation experiment, you need a model that represents your
population or phenomena of interest and a way to generate random numbers
(according to your model) using a computer. The data that are generated
from your model can then be studied as if they were observations. As we will
see, one can use statistics based on the simulated data (means, medians,
modes, variance, skewness, etc.) to gain understanding about the population.
In Section 6.2, we give a short overview of methods used in classical infer-
ential statistics, covering such topics as hypothesis testing, power, and confi-
dence intervals. The reader who is familiar with these may skip this section.
In Section 6.3, we discuss Monte Carlo simulation methods for hypothesis
testing and for evaluating the performance of the tests. The bootstrap method

© 2002 by Chapman & Hall/CRC