The Geometric Shrinkage Estimation Of Covariance Matrix And Its Applications

Covariance matrix not only depicts the degree of diversion of each variable,but also describes the linear correlation between variables,which occupies an important position in multivariate statistical analysis.For example,in the principal component analysis(or factor analysis),the important principal component(or main factor)is selected according to the eigenvalue of the covariance matrix.In linear discriminant analysis(LDA),discriminant functions contain covariance matrices.In exploratory data analysis and test,the independence and conditional independence of variables are measured by covariance matrix.The covariance matrix is also included in the confidence interval.Therefore,the estimation of covariance matrix is a very important problem.It is well known that the sample covariance matrix is an excellent estimator of the population covariance matrix(such as consistent estimator,unbiased estimator,uni-form estimator,etc.)when the dimension p is very small.However,with the increase of the population dimension p,the sample covariance matrix becomes more and more unstable,and the minimum eigenvalue of the sample covariance matrix is much smaller than that of the population covariance matrix,and the maximum eigenvalue is much larger.Secondly,as the p gradually increases,the parameters to be estimated of the population covariance matrix increase very quickly(the number of the parameters is 0(p2)).The increasing of parameters to be estimated and the high interference infor-mation result in the increasing in estimation error.Thus,as the p increases,the sample covariance matrix is no longer a good estimator of the population covariance matrix.In particular,the sample covariance matrix is singular when the population dimension p is larger than the sample size n,that is p>n(at this time the corresponding co-variance matrix is estimated as the high-dimensional covariance matrix).Generally,in theoretical research and practical applications,we assume that the population co-variance matrix is a positive definite matrix,so it is not appropriate to use a singular matrix as an estimator of a positive definite matrix.In a word,it is of great theoretical and practical significance to find a good estimator of the high dimensional covariance matrix in modern statistics.The estimation of high dimensional population covariance matrix is the core issue in modern statistics,and it is also a very challenging problem.In the past more than 10 years,many scholars have been working to improve the estimation of the high dimen-sional population covariance matrix and put forward many estimation methods.Com-mon methods include regularization methods,shrinkage methods,and some models are introduced to describe the correlation between variables in estimating the population covariance matrix.Although these specific methods have many good properties,there are also obvious flaws.For example,the banding method and the thresholding method cannot maintain the positive definiteness of the matrix.The calculation of the tapering method and the thresholding method for positive definite matrices is very complicated.The regularization method and the shrinkage method may change the structure of the population covariance matrix with prior structure.The introduction of a model to describe the relationship between variables contain some limitations in the application because that the assumption is too strong.On the basis of Tong and Wang(2007),a new method of estimating the population covariance matrix-geometric shrinkage estimation is systematically studied in this paper.The method can unify the banding method,the tapering method,the thresholding method and the arithmetic shrinkage estimation to a uniform framework.At the same time,the final estimator has good properties such as simple calculation,maintaining positive definiteness and maintaining a priori structure.In this paper,we study the geometric shrinkage estimation of the high dimensional population covariance matrix and its application in discriminant analysis and asset portfolio.The details are as follows.The first chapter elaborates the research background of this paper.Second,the research history and present situation of the high dimensional covariance matrix esti-mation are reviewed.Finally,the research contents of this paper are given.The second chapter gives some basic knowledge of this paper,which covers the following three aspects.First,the outline of geometric mean and arithmetic mean is introduced.The second is the concept of simultaneous diagonalization of two matri-ces.Finally,the symbolic representation of Hadamard product and its properties are introduced.The third chapter mainly studies the geometric shrinkage estimator of diagonal covariance matrix ∑ = diag(σ11,· · ·,σPP).Firstly,the optimal shrinkage parameter is deduced under the minimization of Log-Euclidean squared loss function.Secondly,the geometric shrinkage estimator and the limit properties of the optimal shrinkage pa-rameter under two kinds of target matrices are studied.Thirdly,the good performance of the geometric shrinkage estimator is verified by simulation.Finally,an empirical analysis is carried out through the data of infectious diseases.The fourth chapter mainly studies the geometric shrinkage estimates of general positive definite covariance matrix ∑ =(σij)pxp.When p>n,the sample covariance matrix S is no longer a positive definite matrix.So this chapter first adds a small per-turbation to the sample covariance matrix to make it a positive definite matrix.Then,the geometric shrinkage estimator is constructed and the optimal shrinkage parameter is deduced under the minimization of Log-Euclidean squared loss function.Finally,the good performance of geometric shrinkage estimation is verified by simulation.The fifth chapter mainly studies the geometric shrinkage estimates of the definite covariance matrix ∑=(σij)pxp with some given structures.Firstly,the geometric shrinkage estimator of the covariance matrix with structure is proposed under the Hadamard product frame.Secondly,a lot of common target matrices are proposed.Then,the optimal shrinkage parameter is deduced under the minimization of Log-Euclidean squared loss function.Finally,the good performance of geometric shrinkage estimation with structure is verified by simulation.The sixth chapter focuses on the application of the geometric estimators of diago-nal covariance matrices in diagonal discriminant analysis.Firstly,the theory of diago-nal linear discriminant analysis(DLDA)and diagonal quadratic discriminant analysis(DQDA)is introduced.Then the geometric shrinkage estimators of diagonal covari-ance matrices are applied to diagonal linear discriminant analysis(DLDA)and diagonal quadratic discriminant analysis(DQDA),and geometric shrinkage linear discriminant analysis(GDLDA)and geometric shrinkage quadratic discriminant analysis(GDQDA)are obtained.Secondly,the advantages and disadvantages of various methods are com-pared by simulation analysis.Finally,the colon gene data were used to verify the error rate of each discriminant analysis method.Both simulation and empirical results show that GDLDA and GDQDA have the lowest misjudgment rate in most cases.The seventh chapter focuses on the application of the estimator of general positive definite covariance matrix in the asset portfolio.Firstly,the theory of the Global Minimum Variance Portfolio(GMVP)is introduced,and the analytic solution of the optimal weight of the portfolio is obtained by using the geometric shrinkage estimator of the general positive definite covariance matrix.Secondly,using the data of the 2015-2016 Shanghai A shares downloaded by CSMAR,the optimal weights of the portfolios are calculated by the arithmetic shrinkage estimator,the geometric shrinkage estimator and the sample covariance.The income of the equal right portfolio is used as the benchmark value,and the difference between the expected income and the benchmark value of each portfolio is compared.The empirical results show that the geometric shrinkage estimation is more effective in the portfolio.The eighth chapter is the main conclusions and prospects of the full text.The main work,the main conclusions of this paper and the future prospects are given respectively.