p-values from bootstrap - what am I not understanding?

Johan Jackson wrote:

Dear stats experts:
Me and my little brain must be missing something regarding bootstrapping.
I
understand how to get a 95%CI and how to hypothesis test using
bootstrapping
(e.g., reject or not the null). However, I'd also like to get a p-value
from
it, and to me this seems simple, but it seems no-one does what I would
like
to do to get a p-value, which suggests I'm not understanding something.
Rather, it seems that when people want a p-value using resampling methods,
they immediately jump to permutation testing (e.g., destroying
dependencies
so as to create a null distribution). SO - here's my thought on getting a
p-value by bootstrapping. Could someone tell me what is wrong with my
approach? Thanks:

STEPS TO GETTING P-VALUES FROM BOOTSTRAPPING - PROBABLY WRONG:

1) sample B times with replacement, figure out theta* (your statistic of
interest). B is large (> 1000)

2) get the distribution of theta*

3) the mean of theta* is generally near your observed theta. In the same
way
that we use non-centrality parameters in other situations, move the
distribution of theta* such that the distribution is centered around the
value corresponding to your null hypothesis (e.g., make the distribution
have a mean theta = 0)

4) Two methods for finding 2-tailed p-values (assuming here that your
observed theta is above the null value):
Method 1: find the percent of recentered theta*'s that are above your
observed theta. p-value = 2 * this percent
Method 2: find the percent of recentered theta*'s that are above the
absolute value of your observed value. This is your p-value.

So this seems simple. But I can't find people discussing this. So I'm
thinking I'm wrong. Could someone explain where I've gone wrong?

There's nothing particularly wrong about this line of reasoning, or at least
not (much) worse than the calculation of CI. After all, one definition of a
CI at level 1-alpha is that it contains values of theta0 for which the
hypothesis theta=theta0 is accepted at level alpha. (Not the only possible
definition, though.)

The crucial bit in both cases is the assumption of approximate translation
invariance, which holds asymptotically, but maybe not well enough in small
samples.

There are some braintwisters connected with the bootstrap; e.g., if the
bootstrap distribution is skewed to the right, should the CI be skewed to
the right or to the left? The answer is that it cannot be decided based on
the distribution of theta* alone since that depends only on the true theta,
and we need to know what the distribution would have been had a different
theta been the true one.

The point is that these things get tricky, so most people head for the safe
haven of permutation testing, where it is rather more easy to feel that you
know what you are doing.

For a rather different approach, you might want to look into the theory of
empirical likelihood (book by Art Owen, or just Google it).

--
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