贝叶斯离散分位数回归模型：理论，方法及应用

Many random variables in practice take discrete values or count outcomes. Regression analysis based on this type of dependent variables include Logistic regression, ordinal regression, and multinomial regression and Poisson regression and so on. Also, the state space of the hidden variables in the famous Hidden Markov models (HMMs) is discrete. Quantile regression for continuous dependent variable is a standard regression analysis tool and is becoming increasing popular nowadays. Bayesian statistics and Lasso-based variable selection have been hot topics and popular in numerous fields in which statistics is applied in last two decades. Bayesian inference quantile regression has attracted a lot of attention in literature recent ten years. However, there is little research on discrete quantile regression and quantile HMM regression and there is even no research on Bayesian inference these discrete quantile regressing models and HMM quantile regression, but highly expected. Based on our experience of developing Bayesain inference continuous-type of quantile regression models, it's highly timing for us to explore Bayesian inference these discrete quantile regression models and variable selection. In the project, we will make a comprehensive study and deep investigation of Bayesian inference discrete quantile regressing models, including Bayesian HMM quantile regression, from theory, method, algorithm, software to application. Via smoothing quantile for counts with Jittering algorithm, we start both parametric Bayesian inference and nonparametric Bayesian inference the following models and variable selection one by one: logistic regression, ordered probit regression, Poisson regression and HMM quantile regression. Parametric method will combine an asymmetric Laplace distribution based likelihood and Jittering method. Nonparametric method will be based on Dirichlet mixtire priors. We will investigate consistency and asymptotic normality of posterior estimates under misspecification of likelihood fucntion.We will also explore proper posterior properties and prior selection, Bayesian regression variable selection. The algorithms will include Jittering algorithm, MCMC algorithms (both Metropolis-Hastings and Gibbs sampling) and Forward-Backward algorithm for updating hidden states. The application will include Xinjiang demographic count data analysis, employment difficulty analysis in ethnic minorities of Xinjiang, health count data analysis of death and ill from air and traffic pollution in Urumqi and financial risk analysis.

实践中许多随机变量取离散值。以这些变量为因变量的重要回归分析包括Logistic回归，有序回归等。此外,隐马尔可夫模型（HMM）的隐变量的状态空间是离散的。分位数回归或依据因变量分位数的回归已经成为常用的分析工具。贝叶斯统计和以Lasso为基础的变量选择方法是当今的热门课题。近年来贝叶斯推断因变量为连续的分位数回归有很多应用和文章发表。但很少有研究离散分位数回归和HMM分位数回归。用贝叶斯方法推断这些分位数回归几乎是空白，但是它极其重要和具有广阔的应用前景。我们将利用近年在发展贝叶斯推断连续依赖性变量分位数回归的经验优势来发展贝叶斯推断离散分位数和HMM分位数回归，研究后验估计的一致性和渐近正态性,先验选择，参数贝叶斯和非参数贝叶斯，变量选择。研发方法有离散模型连续化，MCMC算法和正倒向隐藏状态更新算法等。应用包括新疆人口数据分析，少数民族就业困难分析,城市机动车导致空气污染状况分析等。

回归分析或回归模型是统计中全面描述变量间关系最有用的工具. 非均值回归(包括分位数回归, 极值回归, 众数回归) 比均值回归具有更广泛的应用，但在方法和实际应用这些回归模型面临挑战。挑战内容包括使用贝叶斯方法进行推断和应用这些回归为离散型数据分析多年间愈来愈引起广泛重视和关注。这个项目是应对这些挑战中的一部分. 主要研究内容包括 (1)离散型数据分析 (例如生活质量或健康数据分析) (2)分位数回归 (3)寿命分布推断 (4)极值分布推断 (5)贝叶斯推断.. 项目组成员一起合作19篇文章 (发表14篇SCI 或 SSCI文章. 五篇文章已提交并正在审议). 获得重要结果包括:.(1) 当数据量小时，经典方法如最大似然估计失效.我们提出一种新的和高效的方法用于统计推断一类非平均正态分布 (Yu, Wang and Patilea: 2013) ;.(2) 当密度回归吸引注意时，我们提出新的密度回归方法(Dang and Yu: 2015);.(3) 当光滑参数选择是非参数分位数回归的关键时，我们提出新的选择方法 (Yu, Dang, Zhu and Al-Hamzawi: 2013);.(4) 小样本极值分布推理 (He, Sheng, Wang and Yu: 2014) ;.(5) 第一次提出贝叶斯离散分位数回归 (Yu and Liu: 2016);.(6) 众数估计方法 (Tian, He and Yu, 2016), 等等..应用研究包括肥胖因素调查 (Yu, Liu, Al-Hamzawi Lord: 2016), 收入分配差距测度方法 (田茂茜；万亮；虞克明, 2015), 生活质量或健康数据分析 (Yu and Tian: 2016; Tian and Yu, 2016), 银行业如何影响房地产业 (田茂茜, 虞克明, 2015) , 等等..项目负责人于2013在德国统计周作分位数回归主题演讲. 这个一周会议是世界大的统计会议之一. 由德国统计学会, 德国人口学会和意大利统计学会协会主办。见（http://www.statistische-woche.de/fileadmin/Guide_berlin.pdf）。 项目负责人还代表项目组在2013年2月日本神户大学SESAMI计划会议. 他介绍本项目. 文章和演讲都得到广泛关注。