Mean Pythagorean Relations in the Estimation of the Multi-dimensional Parameter

库尔巴克-莱布勒损失现在甚至在信息论和量子力学中也是重要的。本研究的目的是通过我们对概率分布空间结构的更深入的了解来开发新的方法。本文的研究内容包括：1)条件似然方法和边际似然方法，2)经验贝叶斯方法，3)估计方程方法。本文的研究成果概括如下：1)通过对偶概念探讨参数正交性。这项研究使我们对GLM的基础有了更深入的了解，也为我们提供了一个关于LRT的新发现。提出了参数正交性与估计函数理论之间的密切联系。2)参数正交性回归方法与非参数回归方法之间存在着很大的差距。然而，人们相信，各种共同的性质仍然有效。这一猜想通过模拟研究得到了肯定的证实。因此，下一步将是探索发现的理论背景。3)预计最大似然估计的实质性缺陷是其潜在的过度拟合。这一点在与Ihara教授(大阪E.I.U)一起进行的因子分析中得到了验证。基础性研究涉及到通过Kullback-Leibler损失来研究MLE的渐近二阶结构，以及关于正交参数坐标的研究。尽管这些主题很重要，但对它们的研究似乎很少。江口教授(ISM)和山本教授(美国冈山)积极参与这些联合研究。

The Kullback-Leibler loss is now important even in information theory and quantum mechanics. The aim of the present research is to develop new methods through our deeper understanding of the structure of the apace of probability distributions. The targeted areas are : 1) The conditional likelihood method and the marginal likelihood method, 2) The empirical Bays method, and 3) The estimating equation method.The finding obtained as the present research project are summerized as follow :1) Parameter orthogonality is explored through the notion of duality. This study brings us to deeper understanding of the foundation of GLM.It provides us also with a new finding of the LRT.Recent discussions with Dr.Kawanabe (Tokyo U.) suggest a close relation between parameter orthogonality and the theory of estimating functions.2) There is a severe gap between the paramentric and the nonparamentric regression methods. However, it is believed that various common properties still hold. This conjecture is exploited affirmatively through simulation studies. Thus the next step will be to explore theoretical backgrounds of the findings.3) A substantial defect of the MLE is expected as its potential overfitting. This is examined in the factor analysis, which was conducted with Prof.Ihara (Osaka E.I.U.).Foundamental researches pertain to the asymptotic second order structure of the MLE through the Kullback-Leibler loss and also to orthognal parameter coordinats. Researches on these subjects look sparse in spite of their importance. Professors Eguchi (ISM) and Yamamoto (Okayama U.S.) participated agressively to these joint researches.