基于狄利克雷过程的非参数分位数推断
Nonparametric quantile inference using Dirichlet processes
AI总结：
本文利用狄利克雷过程开展贝叶斯框架下的非参数分位数推断，推导了分位数后验分布等结果，还拓展了相关非参数估计量，为分位数相关统计量的估计提供了新方法。
AI中文摘要：
本章从贝叶斯视角出发，利用狄利克雷过程开展分位数的非参数推断研究。文中刻画了分位数的后验分布，给出了后验均值与方差的显式公式。与分布函数的贝叶斯估计量不同，本文提出的分位数函数的贝叶斯估计量为一条光滑曲线。研究给出了伯恩斯坦-冯·米塞斯型定理，展示了分位数过程的极限后验分布，并建立了其与核平滑分位数估计量的关联。作为副产品，本文还开发了一种自动非参数密度估计量，该估计量无需平滑参数，且支持域与数据范围完全匹配。此外，本文还针对其他分位数相关量给出了非参数贝叶斯估计量，包括洛伦兹曲线、基尼系数、Doksum位移曲线、两样本情形下Parzen的比较分布，以及含协变量时的分位数回归函数。
英文摘要：
This chapter deals with nonparametric inference for quantiles from a Bayesian perspective, using the Dirichlet process. The posterior distribution for quantiles is characterised, enabling also explicit formulae for posterior mean and variance. Unlike the Bayes estimator for the distribution function, our Bayes estimator for the quantile function is a smooth curve. A Bernstein--von Mises type theorem is given, exhibiting the limiting posterior distribution of the quantile process. Links to kernel-smoothed quantile estimators are provided. As a side product we develop an automatic nonparametric density estimator, free of smoothing parameters, with support exactly matching that of the data range. Nonparametric Bayes estimators are also provided for other quantile-related quantities, including the Lorenz curve and the Gini index, for Doksum's shift curve and for Parzen's comparison distribution in two-sample situations, and finally for the quantile regression function in situations with covariates.
