Empirical Likelihood and Small Samples
A Monte Carlo simulation compares 9 methods for setting central 95% confidence intervals for the mean of a small sample, for 7 different sampling distributions. The bootstrap t method is the clear winner in terms of coverage accuracy provided at least 4 observations are available. The confidence intervals tend to be long, but not unreasonably so provided at least 6 observations are available. Thus it is not true, as is commonly supposed, that small samples require one to use parametric methods.
In an extreme case, sampling from the lognormal, the bootstrap t intervals are much longer than those of any of the other methods. The extra length is not excessive, in that other methods do not cover the mean more often than the bootstrap t, when they use intervals of the same length as the bootstrap t.
The coverage levels attained are very close to predictions from asymptotic theory, for n as small as 18, except for very heavy tailed distributions.
An analysis is made of the bootstrap t intervals in small samples. They are seen to be sensitive to small gaps in the order statistics from the sample. Bounds on the sampling distribution of the lengths are found using the gaps.
KeywordsInterval Length Empirical Likelihood Coverage Level Central Interval Coverage Accuracy
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