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

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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.