A Distribution-free Geometric Shrinkage Estimation Of Covariance Matrix

Multivariate statistical analysis plays an important role in modern scientific research.As the core issue of multivariate statistical research,estimation of covariance matrix plays an indispensable role in both theory and practical application.With the emergence of a large number of high-dimensional data,classical covariance matrix estimation methods are no longer applicable.In order to solve the problem of covariance matrix estimation of high-dimensional data,scholars have proposed many new estimation methods.Among them,the methods of shrinkage estimation are very popular,such as the arithmetic shrinkage estimation of Ledoit and Wolf(2004),and the geometric shrinkage estimation of Tong and Wang(2007).In this paper,a generalized asymptotically optimal geometric shrinkage variances estimation is studied under the framework of Riemannian manifold.The application of Riemannian framework gives a new explanation to geometric shrinkage estimation.Thus,the arithmetic shrinkage estimation and geometric shrinkage estimation are calculated in a unified way.At the same time,the estimation of the covariance matrix is not limited by any distribution.The main contents of this paper are as follows.In chapter one,we mainly discuss the background,significance,development process and research status of covariance matrix estimation for high-dimensional data.In chapter two,two asymptotically optimal shrinkage parameter estimates are proposed by using the square loss function of Riemann metric under the framework of Riemann manifold,which gives the shrinkage estimates new explanation and solves the estimation problems without distribution restrictions.In chapter three,through the simulation of normal and non-normal situations,the new estimation method is tested by using the percentage related improvement in average loss(PRIAL)criterion which measures the degree of improvement over the naive sample variance estimators.The results show that the two methods have good performance.Chapter four objectively evaluates and compares the new estimation methods proposed in this paper through the empirical analysis of real data.The fifth chapter is the summary part,we give the main research process and conclusions of this paper.