稳健混合联合位置与尺度模型的参数估计

发布时间：2018-04-26 06:36

本文选题：稳健 + 混合模型　；参考：《昆明理工大学》2017年硕士论文

【摘要】：在统计学中影响统计结果的重要因素有两个:一是观测数据,二是对总体某些特性(分布、独立性等)的假设.当观测数据中存在一些不能很好的代表总体的异常点或者研究总体不满足一些传统的统计方法对总体某些特性的假设时,就会出现问题甚至导致错误的结论.这个时候,一些更为稳健的统计方法、更为稳健的分布类型更能体现出在处理这类问题上的优势,t分布、Laplace分布、Pearson type Ⅶ分布等一些包含异常点的"厚尾分布"对异常值和偏离均值较多的厚尾数据都不是特别敏感,是一种很不错的稳健分布类型,同时也体现出了稳健统计方法的特点:即使存在少量异常点,对与理想分布的偏离所引起的结果影响也不是很大;存在较多的异常点也不至于导致错误的结论.随着社会的发展,我们生活中各个领域的数据也越来越复杂、多样,这时势必要对这些异质的总体进行聚类分析,混合模型应运而生,用不同的参数和比例的分布来拟合不同的几类数据.大量异方差数据的存在违背了传统回归模型中方差齐次性的假设,为了有效的控制方差,在处理异方差数据的问题上,我们多采用联合均值与方差模型,现在我们也可以将模型方法进行推广,使适用范围更加广泛,把同质总体中的联合均值与方差模型推广到异质总体的混合模型中.进一步地,当考虑混合数据的分类情况未知时,我们还可以引入混合专家系统,对混合比例进行建模,应用Logistic回归对影响混合比例的未知参数进行估计.本文主要基于t分布、Laplace分布、Pearson type Ⅶ分布三种稳健的分布应用EM算法对异质总体的混合联合位置与尺度模型的未知参数进行极大似然估计,主要内容有:第一,基于t分布下,建立混合联合位置与尺度参数的模型,应用EM算法、极大似然估计、Gauss-Newton迭代算法对模型中的未知参数进行估计,并通过Monte Carlo模拟方法验证所提出估计方法的有效性.然后试着把所提出的估计方法与实际生活联系起来,解决一些实际问题.第二,基于Laplace分布下,建立混合联合位置与尺度参数的模型,应用EM算法、极大似然估计、Gauss-Newton迭代算法对模型中的未知参数进行估计,并通过Monte Carlo模拟方法验证所提出估计方法的有效性.然后试着把所提出的估计方法与实际生活联系起来,解决一些实际问题.第三,基于Pearson type Ⅶ分布下,建立混合联合位置与尺度参数的模型,.应用EM算法、极大似然估计、Gauss-Newton迭代算法对模型中的未知参数进行估计,并通过Monte Carlo模拟方法验证所提出估计方法的有效性.然后试着把所提出的估计方法与实际生活联系起来,解决一些实际问题.第四,基于Laplace分布下,在混合专家系统中,建立混合联合位置与尺度参数的模型,应用MM算法、EM算法、极大似然估计、Gauss-Newton迭代算法对模型中的未知参数进行估计,并通过Monte Carlo模拟方法验证所提出估计方法的有效性.然后试着把所提出的估计方法与实际生活联系起来,解决一些实际问题.
[Abstract]:There are two important factors affecting statistical results in Statistics: one is the observation data, and the two is the assumption of the general characteristics (distribution, independence, etc.). When there are some exceptions in the observation data that can not be well represented as a whole, or the study generally does not meet the assumptions of some traditional statistical methods on some of the overall characteristics of the general statistical method, At this time, some more robust statistical methods, the more robust distribution types can reflect the advantages of dealing with these problems, t distribution, Laplace distribution, Pearson type VII distribution and other "thick tail distribution" containing abnormal points, and the heavy tailed data with more deviations from the mean value. It is not particularly sensitive, it is a very good robust distribution type, and it also embodies the characteristics of robust statistical methods: even if there is a small number of anomaly points, the effects on the deviations from the ideal distribution are not very large; there are many anomalies that do not lead to the wrong conclusions. With the development of society, we live in our lives. The data in various fields are becoming more and more complex and diverse. At this time, we must cluster analysis of these heterogeneous groups. The mixed model comes into being and fits different kinds of data with different parameters and proportions. The existence of a large number of heteroscedasticity data is contrary to the hypothesis of the traditional Chinese homogeneity in the traditional return model, in order to be effective. In the control of variance, we use the combined mean and variance model in dealing with the problem of heteroscedasticity. Now we can also extend the model method to make the scope of application more extensive, and extend the joint mean and variance model in homogeneity to the mixture model of heterogeneous population. When the class situation is unknown, we can also introduce a hybrid expert system to model the mixture ratio and estimate the unknown parameters that affect the mixture ratio by Logistic regression. This paper is based on three robust distributions of t distribution, Laplace distribution, Pearson type VII distribution and the mixed joint location and scale of EM algorithm for heterogeneous population. The main content of the unknown parameters of the model is maximum likelihood estimation. The main contents are as follows: first, based on the t distribution, the model of mixed joint position and scale parameter is established. EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the Monte Carlo simulation method is used to verify the proposed estimation method. Then, we try to connect the proposed method with real life and solve some practical problems. Second, based on the Laplace distribution, the model of mixed joint position and scale parameter is established. The EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the Monte Carlo module is used. This method verifies the effectiveness of the proposed method. Then we try to connect the proposed method with real life and solve some practical problems. Third, based on the Pearson type VII distribution, a model of mixed joint position and scale parameters is established. The EM algorithm, maximum likelihood estimation, and Gauss-Newton iterative algorithm are applied to the model. The unknown parameters are estimated, and the effectiveness of the proposed method is verified by the Monte Carlo simulation method. Then, we try to connect the proposed method with real life and solve some practical problems. Fourth, based on the Laplace distribution, the mixed joint location and the scale parameter model is established in the mixed expert system. The MM algorithm, EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the effectiveness of the proposed estimation method is verified by the Monte Carlo simulation method. Then, the proposed estimation method is linked to the actual life, and some practical problems are solved.

【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1

【参考文献】