基于经验似然的流形条件Fréchet均值同时置信带与均值集

Manifold Conditional Fréchet Mean Based on Empirical Likelihood, with Simultaneous Confidence Band and Mean Set

摘要: 对流形数据的位置参数，统计推断通常只给出单个点或者逐点意义下的置信区域。然而在许多实际问题中，人们关心的并不是某一个点，而是一族点或者一个集合：例如沿某个协变量连续变化的整条条件均值曲线，又或者当总体位置本身不唯一时由多个点共同组成的集合。对这类对象，逐点区间无法控制整体的置信水平，必须借助同时的或者集合型的推断工具。本文在流形数据的经验似然框架下处理两类这样的问题：一是条件Fréchet均值在有限预测值网格上的同时置信带，二是Fréchet均值不唯一时的均值集置信区域。之所以选用经验似然，是因为其统计量只使用估计函数的矩协方差、并不涉及Fréchet函数的Hessian，因而在Hessian退化时依然有效。文中引言先按流形统计推断、经验似然、条件Fréchet均值与Fréchet均值集四个方向综述已有进展。就方法而言，前者由局部经验似然统计量的逐点极限经联合界得到，不需要带宽收缩下的经验过程Donsker条件；后者则以经验似然函数作为准则函数构造水平集估计，并用乘子自助校准其覆盖概率。文中还说明了乘子自助阈值在此设定下渐近正确的理由，并讨论了网格点数随样本量增长时所面临的技术困难。数值研究的结果与理论一致，文中最后讨论了向连续预测带推广的方向。

Abstract: For the positional parameters of manifold data, statistical inference usually only provides confidence regions at a single point or point by point sense. However, in many practical problems, people are not concerned with a single point, but with a family of points or a set: for example, the entire conditional mean curve that continuously changes along a covariate, or a set composed of multiple points when the population position itself is not unique. For such objects, point by point intervals cannot control the overall confidence level, and simultaneous or set based inference tools must be used. This article deals with two types of problems in the empirical likelihood framework of manifold data: one is the simultaneous confidence band for the conditional Fréchet mean on a finite prediction-value grid; the other is the confidence region for the mean set when Fréchet means are not unique. The reason for choosing empirical likelihood is that its statistic only uses the moment covariance of the estimating function and does not involve the Hessian of the Fréchet function, so it is still effective in when the Hessian is degenerate. The introduction of the article first summarizes the progress in manifold statistical inference, empirical likelihood, conditional Fréchet mean, and Fréchet mean set. In terms of methodology, the former is obtained by the joint bound of the point by point limit of the local empirical likelihood statistic, and does not require the Donsker condition of the empirical process under bandwidth contraction; The latter uses empirical likelihood function as the criterion function to construct level set estimation, and self calibrates its coverage probability using multipliers. The article also explains the reason for the asymptotic correctness of the multiplier bootstrap threshold under this setting, and discusses the technical difficulties faced when the number of grid points increases with the sample size. The results of numerical research are consistent with theory, and the direction of extending to continuous prediction bands is discussed in the end of the article.