2000 Fiscal Year Final Research Report Summary
Integralgeometric distribution theory of random field and its applications to multivariate analysis
Project/Area Number 
11680335

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Statistical science

Research Institution  The Institute of Statistical Mathematics 
Principal Investigator 
KURIKI Satoshi The Institute of Statistical Mathematics, Associate professor > 統計数理研究所, 統計基礎研究系, 助教授 (90195545)

Project Period (FY) 
1999 – 2000

Keywords  stochastic geometry / singularity / likelihood ratio test / categorical data / shrinkage estimation 
Research Abstract 
A realvalued 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 integralgeometric 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 nonregular 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 nonregular 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 twoway ordered categorical data. As a model for twoway 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 integralgeometric 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.

Research Products
(8 results)