Study On Empirical Research And The Relative Efficiency In Parameter Estimation For Linear Model

Linear regression analysis may be the most widely used branch in the statistical analysis. It can back track to the least squares techniques proposed by Gauss in the 19th century, and yielded the regression analysis in the beginning of the 20th century. Parameter estimation in linear model is one of the basic statistical inference forms as well as one of the important research fields in statistics. It has enjoyed many applications in the fields such as biology, chemical industry, medicine, meteorological phenomena, finance, management, industrial technology, agriculture, economic planning and national defence technology. The famous English statistician R.A.Fish has laid a foundation for the parameter estimation theory in the 1920s, and it has made great progress until today, influenced by the A.Wald's statistical decision theory which put forward many new research topics for the parameter estimation theory on the one hand; on the other hand, the development of limit theory has provided the necessary conditions for the development of the large sample theory for estimation.Nowadays, parameter estimation is still a live branch in statistics, and there exist many important estimators such as the least squares estimator, Bayes estimator, ridge estimator, general ridge estimator and James-Stein shrinkage estimator. It's well known that the running of the financial market has become more and more complicated in the last decades, so it's quite necessary to apply the mathematic methods to analysis and control the risk and income on investment. It's a quite phenomenon that there are differences between the theory results and the practices when dealing with the estimation demonstrations, which requires us to make proper adjustments to parameter estimation.In some practical applications of parameter estimation, we are often faced the complicated calculations, which leads to lower efficiency. Furthermore, it's quite often that there are still unknown parameters in the estimators which may lead to the two-stage estimators. However, we still have got little about the statistical properties, especially for the small sample properties about the two-stage estimators. So sometimes we need to carry out statistical inference through replacing the true estimators by some other estimators. A natural problem is that we are to find out the efficiency of such substitutions, which leads to another branch in statistics, namely the relative efficiency of parameter estimation in linear model, where the study on the upper bound of the relative efficiency is a hot problem since it reflects the loss degree of the replacement. Bloomfiled and Watson, Liu, A.Y and Wang, S.G, Huang, Y.L and Chen, J.G has proposed some kinds of relative efficiency with their corresponding lower bounds, respectively.In this paper, we firstly give the introduction and some preliminary knowledges in the 1st and 2nd section, respectively, then in the 3rd part, we make a demonstration about the CAPM's application in the Shen Zhen stock market with the linear regression analysis methods utilizing the SPSS 12.0 software. After that, we discuss some common relative efficiency and their lower bounds in the 4th section. In this part, we also proposed a new kind of relative efficiency and study its lower bound. In the 5th section, we compare some common relative efficiency from both the computation sensitivity and the efficiency points of view, and study the superiority between our new relative efficiency and other common relative efficiency.