复杂多元数据的半参数统计推断

Apart from the discrete and continuous data, there exist various complicated multivariate response data in practice, for example, ordinal categorical data and ranking data that are often-used in psychometric, educational, biological, medical and economical sciences. Based on the single-index dimension-reduction technique, we will develop a series of semi-parametric multivariate statistical models and methods for analyzing those multivariate data in the proposed project. In detail, we will adopt multivariate single-index-type models, multivariate single-index-type probit models and generalized multivariate single-index-type models, to analyze the multivariate continuous responses, the multivariate ordinal categorical data and other data including the ranking data, respectively. Besides those, we will also develop single-index-type structural equation models to model the covariance structure or the conditional covariance structure of those complicated multivariate data. There are at least two innovations in the proposed project: 1) We not only extend the single-index-type models from univariate response to multivariate responses, but also introduce the single-index technique to the research area of structural equation models; 2) In additional to the case that all of the covariates in the indices are the observable variables, we also consider the case that the unobservable (latent) variable(s) corresponding to ordinal categorical and ranking variable(s) acts as the covariate(s) to satisfy the practice needs. For the latter, we first investigate the identifiability conditions of the models. We will make use of Bayesian approach of free-knot splines (i.e., the nonparametric functions in the model are approximated by splines but the numbers and locations of knots are treated as random variables) to make inference about the proposed models via sampling from the joint posterior. The advantage of Bayesian method is that a multivariate response model can be converted into a series of univariate response models by deriving the fully conditional posteriors, which greatly facilitate the sequent analysis. To obtain efficient and fast-convergent algorithms, we will apply acceleration techniques such as generalized Gibbs sampler, alternating subspace-spanning resampling, and partially collapsed Gibbs sampler to our methods of analysis. The research results of the proposed project will lay a solid foundation for the methodological research on the analysis of the complicated multivariate response data.

实际中存在大量的多元响应数据。除了离散型和连续性外，还有各种复杂的数据类型，如心理学、教育学、生物医药和经济中常用的有序分类数据和排名数据等。基于单指标降维技术，本项目将通过发展一系列多元半参数模型和方法来分析这些复杂数据。本项目的创新之处至少包括两点：1)不仅把单指标降维技术应用于复杂多元响应数据的条件均值结构的建模上，还将应用于复杂多元响应数据的协方差结构和条件协方差结构的建模上；2)进入指标的解释变量不仅为可观测的情形，还包括不可直接观测的情形，对于后者，首先解决模型的可识别性问题。本项目将采用自由节点的Bayes方法对建议的模型进行全面的统计分析，包括估计、检验和模型选择等。为构造高效收敛的算法，还将采用一些加速技术。通过本项目的实施，将为分析复杂多元数据提供半参数统计理论和方法。

本项目充分利用非/半参数技术分析了实际中出现的各种复杂数据，包括有序分类数据、排名数据、保险数据、生命和医学数据以及非独立样本数据等等。提出了多元部分线性单指标模型、多元部分线性单指标Probit模型以及潜在单指标模型以分别建模和分析多元连续型响应数据、多元有序分类响应数据和自变量和响应变量均为有序分类变量的数据。为了从含有异常点和异方差性的部分排名数据中聚成稳健和可靠的全排名，在潜在效应框架下，提出了潜在的非对称Laplace模型。利用经验似然方法研究了生存分析中的带有右删失的长度偏差数据，提出了一种直接计算平均剩余寿命的似然比统计量。利用潜在正态模型讨论了医学临床中治疗方案的疗效之间的两两比较问题。提出了一种带有聚类功能的广义期望极大化算法，用于保险损失数据中常用的混合Erlang分布中参数的估计计算。研究了时间序列分析中自回归函数系数滑动平均模型的参数和非参数的Bayes估计和预测问题，证明了鞅差分误差分布下非参数回归模型估计的相合性以及负超可加相依随机变量的加权和的强大数律。本项目完成了14篇学术论文，已正式发表12篇（包含一篇在线发表），其中10篇发表在SCI刊物上,；培养了1名博士研究生和13名硕士研究生。