Hello everybody,
thanks to the numerous answers to my question on bootstrap confidence
intervals (CIs), especially to the people who gave me the reference
"Davison and Hinkley, Bootstrap Methods and Their Application". There I
found the answers to my question. In fact, I had no mistake in my
considerations, there are just different ways to calculate CIs via
bootstrap. These are, to summarize:
1. Bootstrap percentile interval.
This is the method as proposed by Efron (see "DiCiccio and Efron,
Bootstrap confidence intervals, Statistical Science, 1996"). It can, as
far as I understand, be viewed as a non-parametrical refinement of the
standard error. In Efrons method, the confidence limits of a parameter
gamma are simply estimated as respective percentiles of the bootstrap
histogram. This method is appropriate, if the data points, from which
gamma is estimated, obey a symmetric distribution (Efron, however, also
proposes refinements of the method, "bias-correction" and
"acceleration").
2. Basic bootstrap confidence limits
This is basically what I considered, namely to estimate CIs from the
empirical distribution of the "pivot quantity" gamma-gamm0 (where gamma
denotes the estimate obtained from a sample x=(x1...xn), and gamma0 the
true value). In my applications, however,this method did not always
yield reasonable results. The reason may be that gamma-gamma0 is not
really pivotal in these cases, i.e., it still depends on unknown
parameters. Here method 3 might produce better results.
3. Studentized bootstrap confidence limits
Here the empirical distribution of the quantity (gamma-gamma0)/v, where
v estimates the variance of estimates gamma for samples of size n.
Volker Knecht
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