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

现在，即使在信息论和量子力学中，Kullback-Leibler损失也很重要。目前研究的目的是通过我们对概率分布空间结构的更深入的理解来开发新的方法。目标范围是：1)条件似然法和边际似然法，2)经验bayes法，3)估计方程法。本课题的研究成果总结如下：1)通过对偶的概念探讨了参数正交性。这项研究使我们对GLM的基础有了更深的认识。它也为我们提供了一个关于轻轨的新发现。最近与kawanabe博士（东京大学）的讨论表明参数正交性与估计函数理论之间存在密切关系。2)参数回归方法与非参数回归方法之间存在严重的差距。然而，人们相信它的各种共同属性仍然存在。这一猜想通过模拟研究得到了肯定的利用。因此，下一步将是探索这些发现的理论背景。3) MLE可能存在过拟合的缺陷。这在ihara教授（大阪E.I.U.）进行的因素分析中得到了检验。通过Kullback-Leibler损失对MLE的渐近二阶结构和正交参数坐标进行了基础研究。尽管这些课题很重要，但它们的研究却很少。江口教授（ISM）和山本教授（美国冈山）积极参与了这些共同研究。

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.