The theory of the parametric bootstrap is quite similar to that of the
nonparametric bootstrap, the only difference is that instead of simulating
bootstrap samples that are i. i. d. from the empirical distribution
(the nonparametric estimate of the distribution of the data) we simulate
bootstrap samples that are i. i. d. from the estimated parametric model.
All the same considerations arise.
Since the parameter estimate theta hat is not the true
parameter value theta. We do not sample from the correct distribution.
We should sample from Fθ. We do sample with the
same thing with a hat on the θ (which I can't do on a web page).
Thus the bootstrap does not do the right thing, only close to the right
thing when the sample size is large.
In constructing confidence intervals, it helps to bootstrap pivotal
or at least variance-stabilized quantities.
And so forth.
Simulating from a parametric model is not so easy as simulating from the
empirical distribution. In fact, it can be arbitrarily complicated. So
hard that it is an open research problem how to do it. For some
parametric models sampling is easy, others not. In general, it bears no
relation to samping from the empirical.
If the observed data are in the vector x, then
x.star <- sample(x, replace = TRUE)
makes a nonparametric bootstrap sample.
In contrast, if the observed data are assumed to be i. i. d. normal, then
x.star <- rnorm(length(x), mean = mean(x), sd = sd(x))
makes a parametric bootstrap sample. This does not do the right thing
because we should specify mu to be the true population mean
and sd to be the true population standard deviation (but since
we don't know the population values we must use estimates).
For more contrast, if the observed data are assumed to be i. i. d. Cauchy,
then
makes a parametric bootstrap sample. We can't use mean(x)
and sd(x) as estimators of location and scale because the
Cauchy distribution doesn't have moments and hence these aren't consistent
estimators (of anything, much less location and scale).
Why median(x) and IQR(x) / 2 are consistent
(even asymptotically normal) estimators of location and scale would be
more theory than we want to go into here. The only point we wanted to
make is that the three examples look a lot different from each other.
To get to some examples with wading through a tremendous amount of theory,
we will stick to one parametric model for which the sampling looks fairly
similar to the nonparametric bootstrap. This is the multinomial distribution.
The multinomial distribution is the distribution of categorical measurements
on i. i. d. individuals. The number of individuals in each category make up
the data vector x and the probabilities of individuals being
in each category make up a probability vector p
(where probability vector means all(p >= 0) and
sum(p) == 1).
Given a probability vector p of length k
and a sample size n
one creates a multinomial sample with the R statements
(The first statement creates an i. i. d. sample of category numbers with
the specified probabilities. The second counts the number of individuals
in each category. So x.star is a vector of length k.)
For example, suppose we observe the multinomial data defined to
be x in the form below, and we want to test the null hypothesis
that the true category probabilities are all equal (to 1 / 6 because there
are 6 categories). The R function chisq.test does the usual
chi-square test that uses the large-sample approximation (that the chi-square
test statistic has a chi-square distribution). The remainder of the code
does the parametric bootstrap test.
Actually, since the null hypothesis
is completely specified here this is, strictly speaking, a Monte Carlo
test rather than a parametric bootstrap. The test is exact.
For another example, suppose we observe the contingency table defined to
be x in the form below, and we want to test the null hypothesis
of independence (that the row category labels and column category labels
are independent random variables).