Integral-geometric distribution theory of random field and its applications to multivariate analysis

随机场积分几何分布理论及其在多元分析中的应用

• 批准号: 11680335
• 负责人: KURIKI Satoshi
• 金额: $ 1.41万
• 依托单位: The Institute of Statistical Mathematics
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11680335/
• 关键词: stochastic geometry; singularity; likelihood ratio test; categorical data; shrinkage estimation; 尤度比検定

A real-valued random variable with multidimensional indices is called random field. In this research we studied distribution theory of maxima of continuous random field and its applications to statistical inference including multivariate analysis. The distribution of the maxima can be obtained as upper tail probabilities via integral-geometric approach such as tube method and Euler characteristic method. We treat two cases ; one is the regular case where the index set is a closed smooth manifold, and the other is a non-regular case where the index set contains some singularities. The latter is more difficult to treat than the former. However we showed that if the index set is locally convex, the latter non-regular case can be treated similarly to the regular case.As an application to multivariate analysis, we derived the limiting null distribution of likelihood ratio test statistic for testing independence in two-way ordered categorical data. As a model for two-way categorical data where row and/or column categories are ordered, corresponding analysis models with order restricted row and/or column scores are proposed over and over again. In this model we derived an asymptotic expansion for limiting null distribution accurate enough for practical use. Also we provided computer programs to calculate the tail probabilities.Moreover, the integral-geometric method or the tube method which we use in studying the distribution of maxima, is turn to be useful in studying statistical decision theory or estimation theory. We constructed shrinkage estimators towards hypersurface and convex body, where the rate of shrinkage is determined by the curvature of projection onto the surface of convex body.

具有多维指标的实值随机变量称为随机场。本文研究了连续随机场极大值的分布理论及其在统计推断包括多元分析中的应用。最大值的分布可以通过积分几何方法（如管法和欧拉特征线法）获得上尾概率。我们处理两种情况;一种是正则情形，其中指标集是闭光滑流形，另一种是非正则情形，其中指标集包含一些奇点。后者比前者更难治疗。然而，我们证明了，如果指数集是局部凸的，后者的非正则情况下，可以类似于正常的情况下处理.作为多元分析的应用，我们得到了似然比检验统计量的极限零分布，用于测试双向有序分类数据的独立性.作为行和/或列类别有序的双向分类数据的模型，相应的分析模型与顺序限制行和/或列得分被一次又一次地提出。在这个模型中，我们导出了一个渐近展开的极限零分布足够精确的实际应用。此外，我们在研究极大值分布时所采用的积分几何方法或管方法，在研究统计决策理论或估计理论时也是有用的。我们构造了超曲面和凸体的收缩估计，其中收缩率由凸体曲面上投影的曲率决定。

项目成果

Satoshi Kuriki and Akimichi Takemura: "Tail probabilities of the maxima of multilinear forms and their applications"Annals of Statistics. (in press).

Satoshi Kuriki 和 Akimichi Takemura：“多线性形式最大值的尾部概率及其应用”统计年鉴。

Akimichi Takemura and Satoshi Kuriki: "Shrinkage to smooth non-convex cone : principal component analysis as Stein estimation."Communications in Statistics : Theory and Methods. 28・3&4. 651-669 (1999)

Akimichi Takemura 和 Satoshi Kuriki：“平滑非凸锥体的收缩：斯坦因估计的主成分分析。”统计通讯：理论与方法 28・3&4（1999）。

Satoshi Kuriki and Akimichi Takemura: "Some geometry of the cone of nonnegative definite matrices and weights of associated χ^^<-2> distribution."Annals of the Institute of Statistical Mathematics. 52・1. 1-14 (2000)

Satoshi Kuriki 和 Akimichi Takemura：“非负定矩阵锥体的一些几何结构和相关 χ^^<-2> 分布的权重。”统计数学研究所年鉴 52・1 (2000)。

Satoshi Kuriki and Akimichi Takemura: "Shrinkage estimation towards a closed convex set with a smooth boundary"Journal of Multivariate Analysis. Vol.75, No.1. 79-111 (2000)

Satoshi Kuriki 和 Akimichi Takemura：“对具有平滑边界的闭凸集的收缩估计”多元分析杂志。

Satoshi Kuriki and Akimichi Takemura: "Some geometry of the cone of nonnegative definite matrices and weights of associated chi-bar-squared distribution"Annals of the Institute of Statistical Mathematics. 52・1. (2000)

Satoshi Kuriki 和 Akimichi Takemura：“非负定矩阵锥体的一些几何形状和相关卡方平方分布的权重”统计数学研究所年鉴 52・1。