How close are alternative bootstrap P-values?

Chris Lloyd, Melbourne Business School

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This paper has appeared in Statistics and Probability Letters (2011) Volume 80, 1972-1976.

Abstract

The bootstrap P-value based on a test statistic T is the exact tail probability of the observed value t, with the null maximum likelihood estimate of the nuisance parameters substituted. This P-value is know to lead to tests whose size is closer to nominal asymptotically than the first order test. One issue that has not been addressed is whether bootstrap might reduce the impact of the choice of basic test statistic. It is shown that bootstrap P-values based on different first order statistics differ to second order i.e to O(1/m) where m is a measure of sample size. Just as importantly, it is shown numerically that this asymptotic rate can under-estimate how close alternative bootstrap P-values are for small sample sizes. Thsi paper was published in Statistics and Probability Letters (2011) Volume 80, 1972-1976.