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Bootstrap confidence intervals
Worked exampleThe following examples all employ the same statistic, a 10% trimmed mean, and the same data set - the number of larval cryptolignacae upon each of 50 randomly-selected Wobbiewrot's Rattus anilofilous.
The normal approximation confidence limits we obtained were 6.753229 and 7.696771
The studentized mean we obtained was 35.24868
Its percentile limits were 19.61014 and 65.89138
To make those limits comparable to the observed trimmed mean (7.225) we multiplied them by its standard error (the standard deviation of the, first stage, bootstrap trimmed means) giving limits of 4.019533 and 13.505900
The simple percentile smoothed bootstrap limits we obtained were 6.875 and 7.925
Also, to avoid the need for iteration, we obtained our results as a P-value plot and used the values estimated confidence limits by interpolation.
Furthermore because these data were counts, rather than sample an arbitrary (fitted) parametric model, we used a smoothed bootstrap. But, because shifting our model population would change its variance, we calculated the bandwidth after each shift.
Last, since we are shifting the sample but resampling the smoothed sample, assuming the parameter we are tying to estimate is the 10% trimmed mean of the population, we have further assumed that a sample of 50×5000 observations will give us a reasonably unbiased estimate
The test-inversion limits we obtained, after 100 Gaussian-smoothed percentile bootstrap tests, were 6.673416 and 7.668602
The 1-sided P-value plot is below.
Notice that, like ABC limits, these intervals were not constructed assuming is homoscedastic - indeed, given negative data values are rounded to zero, that seemed highly unlikely. Instead this model assumes these data, being right-skewed, have a positive association between location and variance.