Bootstrap Confidence Intervals and Bootstrap Approximations
本文推广了BCa自助法中的变换构造，提出无需自助抽样的新区间BCa₀和分布近似，仅需n+2次统计量计算，适用于复杂参数和非参数问题。
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.