I have generated a nonparametric percentiles bootstrap confidence interval ($32.27143, 51.08571$), and a BCA confidence interval ($33.26, 53.49$) with an initial sample size of $n=7$. Evidently, the BCA interval is larger than the percentiles interval. Shouldn't the BCA interval be better than the percentiles interval (and by "better" I mean "smaller")? I noticed also that when the sample size is larger, for example $n=40$, BCa confidence interval decreases.
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2$\begingroup$ BCa intervals are "better" than other types of bootstrap intervals in terms of coverage. This is, their coverage is closer to the nominal value. In some scenarios this implies that the intervals are wider or narrower than other types of bootstrap intervals. Regarding your last comment, the length of confidence intervals converges to $0$ as the sample size $N\rightarrow\infty$. Intuitively this happens because the larger the sample is, the more information it carries about the parameters and this is reflected on the "precision" of the estimation. $\endgroup$– user10525Nov 15, 2012 at 12:06
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$\begingroup$ Following on Procrastinator's comment, one might think if the intervals were not much different it would be OK to just use the percentile interval. But a habit of doing that will get you that poorer coverage - the intervals do differ in distribution but that does not imply anything about individual instances. $\endgroup$– phaneronNov 15, 2012 at 19:49
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Before using standard bootstrap methods, the data should be tested for bias and skewness beforehand and if they exist, "tricks" should be used to transform the data to correct for this. BCa (bias-corrected and accelerated) bootstrap by Elfron (1987) takes care of these tricks for you.
So, if you have a distribution that has skew and/or bias, you would be able to calculate the true confidence interval. If you randomly drew samples from this distribution and then estimated the confidence interval using the percentile method and the BCa method, the BCa method will tend to be closer to the true confidence interval than the percentile method. The percentile method may worse by underestimating or overestimating the confidence interval. What really matters (although maybe not in the experimenter's mind) is how close the estimate is to the true confidence interval.
As mentioned by @phaneron, you should not simply pick the method that gives you the smaller confidence interval. Doing this would systematically underestimate the confidence interval and make the results invalid.
