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

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的双向有序分类数据的独立性测试。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。

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.