Parametric Bootstrap
If
ˆ
θ
n
is an estimator from an iid sample from a parametric model, it has a sampling
distribution.
If we know the true value of the parameter, say
θ
, then we can find the
distribution of
ˆ
θ
n
. This is because it is a statistic, so it is a function of the data. For the
purpose of our notation, we thus have
ˆ
θ
n
=
h
(
X
1
, . . . , X
n
)
for the appropriate function
h
. We then obtain the distribution of
ˆ
θ
n
using the methods
of transformations.
If the methods of transformations are not easy or useful to use, we can use a Monte
Carlo simulation method to approximate the distribution of
ˆ
θ
provided we also know
θ
0
.
This is the method we used to approximate the sampling distribution in one of our earlier
examples for an iid exponential example.
What happens if we do not know
θ
? The simplest way to implement the Monte Carlo
method is to use a good guess of
θ
0
, in particular
ˆ
θ
n
. This method does work provided the
distribution of the statistic is
continuous
wrt
θ
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 Spring '11
 qqqq
 Statistics, Normal Distribution, parametric bootstrap

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