One-Sided Inference about Functionals of a Density
密度泛函的单侧推断AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要研究一般密度泛函的非参数推断，未涉及所列分位数或分布效应等具体计量方向。
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密度泛函的单侧推断
- 研究问题
- 对未知密度的峰数、模型项数及最优带宽等泛函，在双侧非参数推断通常不可行时能否构造单侧保证？
- 方法与思路
- 作者利用泛函在分布无关度量诱导拓扑下的半连续性，在经验分布的非参数置信邻域内求泛函最小值。
- 材料陈述的贡献
- 摘要称该最小值可提供具有相应覆盖保证的下界，并在显著性水平接近零的高置信情形具有良好渐近性质。
- 阅读关注点
- 应核对双侧推断不可行的准确范围、半连续性的方向、置信邻域构造，以及高置信渐近结论的极限设定。
- 阅读边界
- 仅据版本未注明的索引摘要；不同泛函的适用条件及优化实现未说明，不能将单侧保证推广到任意泛函。
原文摘要
This paper discusses the possibility of truly nonparametric inference about functionals of an unknown density. Examples considered include: discrete functionals, such as the number of modes of a density and the number of terms in the true model; and continuous functionals, such as the optimal bandwidth for kernel density estimates or the widths of confidence intervals for adaptive location estimators. For such functionals it is not generally possible to make two-sided nonparametric confidence statements. However, one-sided nonparametric confidence statements are possible: e.g., "I say with 95% confidence that the underlying distribution has at least three modes." Roughly, this is because the functionals of interest are semicontinuous with respect to the topology induced by a distribution-free metric. Then a neighborhood procedure can be used. The procedure is to find the minimum value of the functional over a neighborhood of the empirical distribution in function space. If this neighborhood is a nonparametric $1 - \alpha$ confidence region for the true distribution, the resulting minimum value lowerbounds the true value with a probability of at least $1 - \alpha$. This lower bound has good asymptotic properties in the high-confidence setting $\alpha$ close to 0.