相依回归模型与扩散过程的统计推断及其应用

项目名称： 相依回归模型与扩散过程的统计推断及其应用

项目编号： No.11471105

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 胡宏昌

作者单位： 湖北师范大学

项目金额： 64万元

中文摘要： 在很多应用中，观测数据通常表现为某种相依性（如：长短相依、混合时间和正负相关序列等），而独立数据的回归模型的研究理论与方法将面临极大挑战，因此本项目将研究相依误差的（部分）线性回归模型的统计推断理论及其在测绘、金融等领域的应用。 一方面，用稳健估计、（拟或Lq）极大似然方法等考虑相依误差的（部分）线性模型，主要研究各种估计量的相合性、渐近分布、收敛速度、重对数律等大样本性质及有关假设检验问题， 使之与独立或短相依误差情形下的对应结果相比较，不仅为测绘、金融等应用领域提供更贴近实际的模型，而且完善回归模型的估计理论和研究方法，试图寻找相依误差回归模型的统一推断理论； 另一方面，由于分形时间序列是一类分数随机微分方程的解，因此利用统计方法研究扩散过程的统计特性，为分形时间序列的研究提供理论基础和新的统计方法，进而初步研究误差为分形时间序列的线性回归模型，开辟回归模型与分数扩散过程新的交叉领域。

中文关键词： 半参数模型；统计推断；参数估计；时间序列分析；扩散过程

英文摘要： In many applications, the observation data are usually dependent,such as long and short dependent, mixing time series, positive and negative correlation sequences,and so on. It will face great challenge for theory and methods of regression model with independent data. Therefore this project will investigate statistical inference of (partially) linear regression models with dependent errors and their applications in surveying and mapping, finance and other fields. On the one hand, we consider the (partially) linear regression with dependent errors by the robust estimation and the (quasi or Lq) maximum likelihood methods. We will mainly study some large sample properties of these estimators,such as consistency and asymptotic distribution, convergence rates, the law of the iterated logarithm, etc. We also study the related hypothesis test problems. These results can be compared with the corresponding results on short dependent or independent errors. The results not only provides the model closer to the actual for surveying and mapping, finance and other applications, but also improve the estimation theory and research methods of regression model.We try to obtain unified inference theory of regression models with dependent errors. on the other hand, because the fractal time series is solution of a fractional random differential equation, we will use statistical method to study the statistical properties of the diffusion process. These results are solid theoretical foundation of fractal time series, and then we may propose new statistical methods.Thus we preliminary study the linear regression model with fractal time series, and open up a new cross field about regression model and fractional diffusion process.

英文关键词： partially linear regression model;statistical inference;parameter estimate;time series analysis;diffusion process