以比值矩陣法處理高維度中相容性問題

Abstract

在給定一些條件分配函數時，若存在聯合分配函數使得其條件分配函數跟給定的條件分配函數相同時，則稱這些給定的條件分配函數是相容的。二維有限離散型相容性問題由Arnold et al.（1989）率先提出以比值矩陣（ratio matrix）作為解決問題的核心工具。Song et al.（2008）延續相關的研究，並提出了IBD比值矩陣（irreducible block diagonal ratio matrix）的突破性概念，給予了是否相容？以及相容時，聯合機率分配是否唯一的簡單、方便、快速檢驗方法。同時，當聯合機率分配函數解不唯一時，Song et al.（2008）也給予了找出所有可能解的方法。本計畫擬將前述二維有限離散的結果推廣至高維度的情形，即給定一些多個隨機變數的完全條件分配函數（full conditional distributions），如何運用相關的比值矩陣，判斷他們是相容的？若相容時，又如何找出所有聯合分配函數的解？

The given conditional densities are said to be compatible if there exists an associated joint density with them as its conditionals. Arnold et al.（1989）first used the ratio matrix to solve for the bivariate case. However, there is some insufficiency in their main theorem. Song et al.（2008）proposed the IBD ratio matrix concept, and gave a simple and easy manipulation method for compatibility checking and uniqueness checking. Besides, they also gave a way to find all possible joint densities whenever the solutions are not unique. In this study, we will extend the results obtained in bivariate case to those in high-dimensional cases for full conditional densities.