New Developments in Nonparametric Bayesian Inference; Univariate and Multivariate time series with infinite varaince.

申请人的研究计划集中在统计推断的两个主要课题上。第一个主题主要涉及贝叶斯推理。在贝叶斯范式中，统计学家认为对一些未知参数具有概率先验知识。之后，收集的数据用于更新所选模型下的先验知识。非参数贝叶斯模型在一个具有最小限制的更一般的框架中考虑先验知识。申请人在这一领域的研究领域主要涉及非参数贝叶斯模型中使用的新过程的构建，以及为这些过程导出快速但更精确的近似工具，以将其用于统计推断。申请人还导出了用于这种过程的有效模拟技术，以便使从业者能够以更快的速度运行他们的计算机程序。 申请人的第二个项目研究了随时间演变的过程的大样本理论。在统计学中，这些过程被称为时间序列。有时，这些时间序列会显示出不寻常的、更高的峰值或极端值。这些尖峰和时间序列中频繁观察到的极端情况是用具有无穷方差的某些随机变量建模的。对于单变量无穷方差时间序列的研究已经相当广泛，而对于多变量无穷方差时间序列的研究却知之甚少。在多变量情况下，每个坐标中的尖峰的大小可以不具有相同的类型和大小。申请人的研究调查了无限方差多变量时间序列的大样本理论。其结果可用于金融、电信等领域的时间序列建模。

The applicant's research program is concentrated on two major subjects in statistical inference. The first subject is mainly related to Bayesian inference. In the Bayesian paradigm, the statisticians considers having a probabilistic prior knowledge on some unknown parameters. Later, the collected data is used to update their prior knowledge under the chosen model. A nonparametric Bayesian model considers the prior knowledge in a much more general framework with minimal restrictions. The applicant's research area in this field is mostly related to the construction of new processes used in nonparametric Bayesian models and the derivation of fast yet more precise approximation tools for these processes to use them in statistical inference. The applicant also derives efficient simulation techniques for such processes in order to enable practitioners to run their computer programs with faster speed. The applicant's second project investigates the large sample theory for processes which evolve in time. In Statistics, these processes are called time series. Sometimes these time series exhibit unusual and higher spikes or extremes. These spikes and the frequent observation of extremes in time series are modelled with certain random variables with infinite variance. There is extensive research for the univariate infinite variance time series but little is known for multivariate infinite variance time series. In the multivariate case, the size of the spikes in each coordinate may not be of the same type and size. The applicant's research investigates the large sample theory for the infinite variance multivariate time series. The result can be used to model time series in Finance, Telecommunications and many other fields.