For most commonly used confidence-interval procedures,
the logic of CI's is exactly the same as the logic of
hypothesis testing. One has a parametric family of
hypotheses with parameter (vector) denoted b, and one
tests not just one, but every possible value of b.
The confidence region (at level 95% for example)
consists of just those values of b that are not
rejected by a .05 level test.
This explanation does not deal with many fundamental
questions--what test statistic to use, what long-run
properties such a procedure has, how to approximate
the distribution of the test statistic, how to
interpret "confidence"--perhaps others. But it
does show that there is nothing wrong with the logic
of CI's that would not also apply to tests of just
a single "null" value b = b0. CI's have the advantage
that they report inferences from the data regarding ALL
values of b, not just some one b0. Many common misuses
of null-hypothesis testing have their absurdities exposed
by use of CI's.
The bootstrap may be regarded as a method of approximating
the distribution of the test statistics for various values
of b. There is a large literature studying conditions
under which such approximations are (usually) good. It
is NOT assumed that the population distribution is
exactly the same as the sample distribution--that would
be absurd. Such an assumption is alluded to only by way
of heuristic explanation of the bootstrap, e.g., "suppose
that we were sampling from a population exactly specified
by this sample distribution, and calculated a certain statistic,
then by simulation we could determine its distribution..."
But that is just heuristic, not assumption, nor yet proof,
and it is only intended as such. I could go on and point
out why this heuristic is quite useful, but perhaps not now.
Dave Krantz (dhk@stat.columbia.edu)