Bootstrap Confidence Intervals and Bootstrap Approximations

Abstract

Abstract The BCa bootstrap procedure (Efron 1987) for constructing parametric and nonparametric confidence intervals is considered. Like the bootstrap, this procedure can be applied to complicated problems in a wide range of situations. For models indexed by a scalar parameter θ with efficient estimator , the BCa procedure relies on the existence of a transformation g(·) such that () is approximately normally distributed with standard deviation 1 + ag(θ), although explicit knowledge of g(·) is not required. In this article, we show how to construct this transformation by generalizing the one given by Efron (1987, sec. 10) for translation families. This construction consists of the composition of a variance-stabilizing transformation and a skewness-reducing transformation. It produces a new interval, the BCa interval, that is asymptotically equivalent to the BCa interval and can be computed without bootstrap sampling. We also derive from this construction an accurate approximation to the bootstrap distribution of that also does not require bootstrap sampling. Both the new interval and the approximation require only n + 2 evaluations of the statistic. Like the BCa procedure, the BCa 0 interval can be extended to multiparameter and nonparametric problems. As an example, we compute the BCa 0 interval for Cox's partial likelihood estimator, a complicated statistic that is obtained by iterative solution of a score equation.