Centralized Parameter Analysis Of Sample Covariance Matrix In Large Dimensional Spiked Model

In the past hundred years of research and development,the broad and profound random matrix theory has gradually penetrated into many disciplines,including nuclear physics,cosmology,number theory,integrable systems,time series analysis,crystallography,quantum chaos,efficient matrix generation,signal processing,image processing,streaming media analysis,transportation,trajectory tracking,neural networks,complex networks,big data and even cultural structure research.Large dimensional random matrix theory is an important research topic of the random matrix in the field of probability and statistics.In recent years,with the rapid development of computer,the amount of data to be processed is increasing.The processing of these data is inseparable from a solid theoretical foundation.Therefore,the research of large-dimensional random matrix has attracted more and more attention.Non mathematical statistics is the most applied branch of mathematics.It is widely used in all walks of life,including the national economy and people’s livelihood,economy,finance and science and technology.Due to the universal application of calculation,most of the data collected by people are of large dimension,even very large dimension.Therefore,it is very important to develop and create a set of large-dimensional data analysis.No matter the national economy and people’s livelihood,science and technology,medicine and health,national defense and military,large dimensional random matrix theory will be well applied.The development of large-dimensional random matrix research and these disciplines have been influencing each other.The expansion of the sample covariance matrix under the large-dimensional framework becomes the large-dimensional sample covariance matrix.In data analysis,many statistics can be expressed by the correlation function of the sample covariance matrix,and the research on its related properties is also evolving and becoming more diverse,but more problems will follow.The generalization of properties in the case of large dimension can not be obtained in the original classical theory alone,and the more data studies have demonstrated this fact.This article is based on the random matrix of large dimensional data,considering some asymptotic properties of linear spectral functions in the centralized sample covariance matrix,the most prominent of which is the application and expansion of the central limit theorem,and considering how to formulate the central parameter in the central limit theorem.When the dimension and sample size are fixed,which central parameter will be more consistent with the estimation situation.Furthermore,in order to specifically represent the central parameter,You can choose the Spiked model as the overall covariance matrix model.Under the Spiked model,you can numerically simulate the estimation of different central parameters.Through comparison,you can know which form of central parameter estimation is closest to the overall data,and thus obtain the desired results.