High-Dimensional Covariance Matrix Estimation In SEMI-Parametric Approximate Factor Model

The estimation of the covariance matrix is the core problem of multivariate statistical studies. In particular, the estimation of the high-dimensional covariance matrix has become a hot area of research owing to the challenge of the theory and the widely applications and the literature about it become more and more in recent decades. For example, Fan, Fan and Lv (2008) combines the estimation of the covariance matrix with the linear model and put forward to estimate the high-dimensional covariance matrix in the linear factor model. On the other hand, researchers found there are some limitations when the linear model is used to de-scribe the relationship between variables with the further research. Thus, many researchers turn their research direction to nonlinear and partially linear model. On these bases, we consider to introduce the semi-parametric approximate factor model to estimate the high-dimensional covariance matrix.The major study of this article is as follows. In chapter one, we discuss the background, meaning and status of the research and introduce the semi-parametric approximate factor model. In chapter two, we use the kernel function method and the least square method to estimate the non-parametric function and the parameter in the above model respectively, then obtain the convergence rate of the estimators. In chapter three, we divide the population covariance matrix into four parts firstly and then choose corresponding methods to estimate each part and obtain the con-vergence rate of the estimator in the end. In chapter four, we use simulation to test the performance of our estimator and compare our estimator with other esti-mators to tell the good from bad. The fifth chapter is about some conclusions and expectations and we put forward some unresolved problems and the future research direction.