The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general distributions in the natural exponential family. The focus of this note is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose using a local asymptotic framework to study this subtle coverage behaviour. Under this framework, we discover that when the true parameters are on, or close to, the restriction boundary, the asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, remarkably, both under and over the nominal level in the two-sample case. Using illustrative examples, we show that the results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems.

当参数受线性不等式约束时，研究了常用的 bootstrap 百分位置信区间的渐近行为。我们专注于从自然指数族中的一般分布生成的数据的重要一样本和二样本问题。本笔记的重点是量化参数自举百分比置信区间的覆盖概率，特别是它们在边界附近的限制行为。我们建议使用局部渐近框架来研究这种微妙的覆盖行为。在这个框架下，我们发现当真实参数在或接近限制边界时，渐近覆盖概率总是可以超过单样本情况下的名义水平；然而，值得注意的是，它们可以是 在双样本情况下低于和高于名义水平。使用说明性示例，我们表明结果为将 bootstrap 百分位数方法应用于约束推理问题提供了理论依据和指导。