MM算法中的几类问题之研究及其应用

The minorization-maximization (MM) algorithm provides an important and useful tool for optimization problems and has a broad range of applications in statistics because of its conceptual simplicity, ease of implementation and numerical stability. A major challenge in constructing an MM algorithm is that one cannot know how to select an appropriate surrogate function. The PI propose to study the following four topics: (Item 1) The assembly-decomposition (AD) MM algorithms; (Item 2) Constructing MM algorithms by using the continuous version of Jensen’s inequality; (Item 3) The one-step-late AD-MM algorithms; (Item 4) The improved De Pierro (DP) algorithm. Item 1 of the project is motivated by the fact that it has to be done case by case for constructing a surrogate function in the existing literature on MM algorithms. Item 1 aims to propose a so-called AD technique which is an efficient and unified method for constructing a surrogate function in a class of MM algorithms; to recommend this method to practitioners. Item 2 of the project is motivated by that in the existing MM literature we cannot find such examples in which the surrogate function was constructed by the continuous version of Jensen’s inequality. Item 2 aims to establish a class of MM algorithms via the continuous version of Jensen’s inequality; to apply them to truncated normal/t distributions and grouped data problems. Item 3 of the project is motivated by that it is extremely difficult to iteratively calculate the maximum penalized likelihood estimates (MPLEs) when the penalty function is quite complicated. Item 3 aims to develop one-step-late AD-MM algorithms; to apply them to a class of complicated models for computing MPLEs or Bayesian estimates or conducting variable selection; to expand the application ranges of the proposed AD-MM algorithms. Item 4 of the project is motivated by four drawbacks in the DP algorithm which is a special member of the family of MM algorithms. Item 3 aims to propose an improved version of the original DP algorithm; to apply it to a class of commonly used models; to enlarge the space of solving practical problems.

本课题拟开展如下四个方面的研究: (项目一)组装分解MM算法; (项目二)利用Jensen不等式的连续版本来构造MM算法; (项目三)一步滞后的组装分解MM算法; (项目四)改进的DP算法。项目一的实际背景来自于: 在存在的MM文献中对替代函数的构造都是个例研究;目标是首次提出组装分解技术,为一类MM算法中的替代函数的构造提供一个统一的方法。项目二的实际背景来自于: 在存在的MM文献中对如何应用Jensen不等式的连续版本来构造替代函数,到目前为止是一个空白;目标是首次提出利用Jensen不等式的连续版本来构造一类MM算法, 并应用到截断分布正态和t分布模型以及分组数据问题。项目三的目标是首次提出一步滞后的组装分解MM算法，并应用到一类复杂模型的极大惩罚似然估计(变量选择)或贝叶斯估计,以扩大组装分解MM算法的适用范围。项目四的目标是提出对传统的DP算法进行改进，并应用到一类常见的模型中。

我们在一类MM算法中, 提出一个称之为组装分解的技术, 来构造相对应的替代函数。如此构建的MM算法简称为组装分解MM算法 (或AD-MM算法)。组装分解(AD)技术由A-技术和 D-技术组成。我们应用所提出的AD-MM算法于一类重要统计模型: 透射层析成象中的泊松模型, 多元复合零浮动广义泊松分布, 多元零壹浮动广义泊松模型, 截断正态分布, 截断学生t模型, 部分非线性模型, 伽玛脆弱生存模型, 一般的不完全分类数据分析, 带三分类缺失数据的不完全列联表, 一类零截断离散分布, 零壹浮动的单纯型回归模型, 二型区间删失数据分析, 混合正态的新型多元Laplace 分布。我们提供了算法的收敛性质, 包括局部收敛, 全局收敛以及收敛率。..该课题研究进展顺利,完全按照研究计划实施, 一共发表了13篇SCI论文, 其中Paper 1是最重要的研究成果。