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

基本信息

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

项目摘要

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.

具有多维指数的实用值随机变量称为随机字段。在这项研究中，我们研究了连续随机场最大值的分布理论及其在统计推断中的应用，包括多元分析。最大值的分布可以通过整数几何方法（例如试管方法和Euler特征方法）作为上尾概率获得。我们治疗两个病例；一个是常规情况，索引集是封闭的平滑歧管，另一个是一个非规范的情况，索引集包含一些奇异性。后者比前者更难治疗。但是，我们表明，如果指数集是局部凸起的，则可以与常规案例相似地处理后一种非规范病例。作为多变量分析的应用，我们得出了可能性比率测试统计量的限制无效分布，以测试在双向有序分类数据中测试独立性。作为订购行和/或列类别的双向分类数据的模型，一遍又一遍地提出了具有顺序限制行和/或列分数的相应分析模型。在此模型中，我们得出了一个渐近扩展，以限制无效分布的准确性以进行实际使用。此外，我们还提供了计算尾巴概率的计算机程序。此外，我们在研究最大值分布中使用的积分几何方法或试管方法，对研究统计决策理论或估计理论非常有用。我们构建了向超表面和凸体的收缩估计量，其中收缩率取决于投影在凸体表面上的曲率确定。