Mathematical Sciences: Smoothing Parameter Selection and Related Topic

这个项目的统计部分将研究一些使用非参数方法的估计方法。特别是核估计和相关领域的非参数和密度估计的应用将进行研究，并解决平滑参数选择问题。要考虑的问题包括不连续或粗糙密度函数、脊回归的选择以及离散化的影响。该项目的建模部分涉及离散时间和连续时间马尔可夫模型的理论和应用的新发展。研究者将集中在收敛的速度结果和国家和历史依赖的标准，这种稳定性。这些结果将被纳入到马尔可夫决策过程的新方法中，其中状态空间被扩展到将决策集包含到整个马尔可夫结构中。然后，在相当复杂的环境中建立这种一般过程的稳定性。概率建模和统计估计技术包含同一活动的两个独立部分。第一部分努力用可管理的术语描述真实且通常复杂系统的模型，使系统中固有的随机性得到适当的表示。第二种尝试是使用这些模型和收集的数据来描述真实的系统，这样人们就可以估计实际描述实际发生的情况的参数。在这个项目的建模部分，研究人员将研究运筹学、时间序列、金融和经济学中的重要模型，重点是开发这些系统稳定性的方法。在提案的统计部分，研究人员将研究一些计算机密集的估计参数的方法，目的是找到快速给出良好估计的方法。

The statistical part of this project will be to study a number of estimation methods using nonparametric approaches. In particular kernel estimation and related areas of nonparametric and density estimation applications will be studied, and the smoothing parameter selection problem addressed. Problems to be considered include those of discontinuities or rough density functions, selection for ridge regression, and the effect of discretization. The modelling part of this project involves new developments of the theory and applications of Markov models in both discrete and continuous time. The investigator will concentrate on rates of convergence results and on state and history dependent criteria for such stability. These results will be incorporated into a new approach to Markov decisions processes where the state space is extended to include the decision set into the overall Markovian structure. Stability properties of such general processes will then be established in quite complex environments. Probabilistic modelling and statistical estimation techniques encompass two separate parts of the same activity. The first part endeavors to describe, in manageable terms, models of real and often complex systems in such a way that the inherent randomness in the system is properly represented. The second endeavors to use these models, and data gathered to describe real systems, in such a way that one can estimate the parameters that actually describe what is happening in practice. In the modelling part of this project the investigators will study important models in operations research, time series, and finance and economics, with a focus on developing methods for stability of such systems. In the statistics part of the proposal, the investigators will look at a number of computer intensive ways of estimating parameters with a goal of finding methods which give good estimates quickly.