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STAT 531: Simulation Study
H. Kim
Department of Mathematics and Statistics
University of Calgary
Fall 2010
H.Kim
1/30
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View Full Document Simulation
A numerical technique for conducting experiments on the computer
Monte Carlo (MC) simulation
: computer experiment involving random
sampling from probability distributions
invaluable in statistics.
. .
usually, when statisticians talk about “simulations,” they mean
“Monte Carlo simulations”
Fall 2010
H.Kim
2/30
Rationale
In statistics
properties of statistical methods must be established so that the
methods may be used with conﬁdence
exact analytical derivations of properties are
rarely
possible
large sample approximations to properties are often possible,
however.
. .
evaluation of the relevance of the approximation to (ﬁnite) sample
sizes likely to be encountered in practice is needed
analytical results may require
assumptions
(e.g., normality), but what
happens when these assumptions are violated
? analytical results, even
large sample ones, may not be possible
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View Full Document Usual Issues
Under various conditions
is an estimator biased
in ﬁnite samples? is it still consistent
under
departures from assumptions? what is its sampling variance
?
how does it compare to competing estimators
on the basis of bias,
precision, etc.?
does a procedure for constructing a conﬁdence interval
for a
parameter achieve the advertised nominal level of coverage
?
does a hypothesis testing procedure
attain the advertised level or size?
if it does, what power
is possible against diﬀerent alternatives to the
null hypothesis? do diﬀerent test procedures deliver diﬀerent power?
How to answer these questions in the absence of analytical results?
Fall 2010
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Monte Carlo Simulation
an estimator or test statistic has a
true sampling distribution
under a
particular set of conditions (ﬁnite sample size, true distribution of the
data, etc.)
ideally, we would want to know this true sampling distribution in
order to address the issues on the previous slide
but derivation of the true sampling distribution is not tractable
⇒
approximate
the
sampling distribution
of an estimator or test statistic
under a particular set of conditions
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View Full Document How to Approximate
A typical Monte Carlo simulation involves the following
generate
S
independent data sets under the conditions of interest
compute the numerical value of the estimator/test statistic
T
(data)
for each data set
⇒
T
1
,...,
T
S
if
S
is large enough, summary statistics
across
T
1
,...,
T
S
should be
good approximations
to the true sampling properties of the
estimator/test statistic under the conditions of interest
e.g., for an estimator for a parameter
θ
:
T
s
is the value of
T
from the
s
th
data set,
s
= 1
,...,
S
,
the
sample mean
over
S
data sets is an
estimate of the true mean
of
the sampling distribution of the estimator
Fall 2010
H.Kim
6/30
Simulations for Properties of Estimators
Simple example:
Compare three estimators for the
mean
μ
of a distribution based on i.i.d.
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This note was uploaded on 02/04/2011 for the course STAT 531 taught by Professor Gaborlukacs during the Spring '11 term at Manitoba.
 Spring '11
 GABORLUKACS
 Statistics

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