Bootstrap-Assisted Inference for Generalized Grenander-type Estimators
广义 Grenander 型估计量的自助法辅助推断AI 解读 ↓
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广义 Grenander 型估计量的自助法辅助推断
- 研究问题
- 对于单调函数的广义 Grenander 型估计量，标准非参数自助法失效且函数平坦程度可能未知时，如何构造可靠推断？
- 方法与思路
- 作者自动变换估计量，并通过数值导数与广义刀切法估计一个标量；在随机抽样下采用可交换自助法实现辅助推断。
- 材料陈述的贡献
- 摘要称，所提程序可适应未知平坦程度，缓解标准自助法不一致问题，并提供相关技术结果、应用例子及小规模模拟。
- 阅读关注点
- 应核对自动变换与标量估计的具体构造、平坦程度适应性的条件，以及可交换自助法如何逼近相应非标准极限分布。
- 阅读边界
- 这是 arXiv 预印本，未核验同行评审或期刊接收；摘要未说明完整正则条件、区间构造及模拟结果，不能将结论直接推广到依赖数据。
原文摘要
Westling and Carone (2020) proposed a framework for studying the large sample distributional properties of generalized Grenander-type estimators, a versatile class of nonparametric estimators of monotone functions. The limiting distribution of those estimators is representable as the left derivative of the greatest convex minorant of a Gaussian process whose monomial mean can be of unknown order (when the degree of flatness of the function of interest is unknown). The standard nonparametric bootstrap is unable to consistently approximate the large sample distribution of the generalized Grenander-type estimators even if the monomial order of the mean is known, making statistical inference a challenging endeavour in applications. To address this inferential problem, we present a bootstrap-assisted inference procedure for generalized Grenander-type estimators. The procedure relies on a carefully crafted, yet automatic, transformation of the estimator. Moreover, our proposed method can be made ``flatness robust'' in the sense that it can be made adaptive to the (possibly unknown) degree of flatness of the function of interest. The method requires only the consistent estimation of a single scalar quantity, for which we propose an automatic procedure based on numerical derivative estimation and the generalized jackknife. Under random sampling, our inference method can be implemented using a computationally attractive exchangeable bootstrap procedure. We illustrate our methods with examples and we also provide a small simulation study. The development of formal results is made possible by some technical results that may be of independent interest.