A New Algorithm Of Biased Estimation In Linear Model

Linear model is an important branch of modern statistics,which plays an important role in the fields of business management,economy,medicine,biology,agriculture,geology,meteorology,society and so on.Parameter estimation is a very important content in the fields of linear model.In the early nineteenth century,Gauss proposed the least squares estimation,which had been proved to be the best linear unbiased estimation.While multicollinearity occurs among the independent variables,which let the least squares estimate become unreliable.In order to improve the least square estimation,James and Stein proposed compression estimation in 1960s,and then more and more statistical scholars had joined in the study of biased estimation and get a lot of influential achievements,which constantly improved the development of parameter estimation in linear model.In this paper,we mainly study the biased estimation of parameter in linear model,introduce and summarize the biased estimation existed and then put forward new algorithm of biased estimation.This thesis was divided into five chapters,in the first two chapters,we present introduction and preliminary knowledge,which respectively introduce the actual background and research status of Biased estimation and some basic theory and relevant knowledge,for example the least square theory,the common biased estimation,Gaussian elimination transform,the ridge trace method to select ridge parameters,the effects of multiple collinearity and the method to diagnose it and so on.In the third and fourth chapters we proposed a new algorithm of biased estimation and used simulation experiment and actual example to evaluate them.In the fifth chapter,we summarized this paper and present the unexhausted questions.The third chapter is the core and key of the thesis.First,conduct Gaussian elimination transform on morbid matrix,we found that when morbidity of matrix occurs,the principle component of the matrix will become very small.Referenced by the method of the ridge estimation which increase the value of whole diagonal elements,we got the inspiration that increase the corresponding principle component before conduct Gaussian elimination transform on it.While we need to solve the problems about how to select the principle component and increment of it before we want to implement new algorithms.In this paper,we put forward two options of the principle,and then put forward two different methods.In the fourth chapter,we carried on data experiment,finally comparing their box plots about the mean residual sum of squares and the mean prediction residual sum of squares to LS estimation and ridge estimation to show that they performance better than these two estimations existed,and then used the second new method to analysis examples of body fat in life.When the multiple collinearity is strong,we found that the first new method performed better than Ridge estimation in terms of forecast effect,while the second new method performs better than Ridge estimation in terms of unbiasedness after carrying on simulation experiments and analysis of actual example.In the fifth chapter,we point out the limitations of both methods and the unexhausted questions.Such as the first new method is applicable to the situation where the sample size is large and we care more the prediction effect.While the second new method is subjective in some sense due to the limitations of the method of ridge trace.