Hello everybody,
could anybody give me a reference (or a hint) to the theoretical
foundation of confidence limits estimated by bootstrap percentiles?
At the first sight, bootstrap percentiles seem a an appropriate method
for confidence estimation, but the closer I look on the method, the
less I understand it.
I tried to obtain a theoretical foundation by viewing the bootstrap as
an empirical pivot method.
For a parameter gamma as estimated by gamma(x) from a sample
x=(x1...xn), the quantity
t'(x', gamma0)=gamma(x')-gamma0 (1)
is a pivot quantity, i.e., its distribution is independent of gamma0. Is
that true?
According to the "Numerical Recipes", the distribution of t'(x',gamma0)
over many real samples x' is approximated by the distribution of
t*(x*)=gamma(x*)-gamma(x) (2)
over many bootstrap replica x*. The bootstrap alpha/2 and 1-alpha/2
percentiles gamma1*,gamma2* for a confidence level 1-alpha then convert
to a confidence interval (t1*,t2*) for t*, which approximates the
confidence interval (t1',t2') for t'. From that, setting x'=x yields a
confidence interval (gamma1(x),gamma2(x)),
gamma_i(x) = 2*gamma(x) - gamma_i*, i=1,2.
This differs from Efrons estimate, and also yields apparently wrong
results when applied to a concrete problem. Where is the error?
Regards,
Volker Knecht
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