On The Equivalence Of Parametric Estimations Under Restricted Models And Transformed Models

Linear statistical model is a very important statistical model.On the one hand,it includes a series of statistical models,such as linear regression model,analysis of variance and covariance analysis model,variance component model and so on.These statistical models have wide and important applications in the development of national economy.On the other hand,the basic theories and methods of linear models also provide basic tools for other statistical theories and methods.Linear statistical model is widely used in modern statistics,and many phenomena in fields such as medicine,economy,meteorology,agriculture and engineering technology can be approximately described by using linear statistical model.At present,the research on linear model mainly focuses on its application and parameter estimation.Ordinary least squares estimator(OLSE),Best Linear Unbiased Estimator(BLUE)are the two most commonly used estimates of parametric functions in linear models.In this paper,from the perspective of algebra and matrix,the generalized inverse of matrix and matrix rank formula are used to study that the relationship between the ordinary least squares estimators and the best linear unbiased estimators of parametric functions under the restricted linear model,the relationship between the ordinary least squares estimators and the best linear unbiased estimators of parametric functions under the transformed linear model and the relationship between OLSE and BLUE of parameter function under restricted model and transformed linear model.This paper will be divided into five parts.The first part of this paper mainly introduces the research background of this paper,research status and significance,as well as the symbolic expression involved in this paper.In the second part,the general linear model,the restricted linear model and the general forms of transformation linear model,as well as some special forms of the transformation model are introduced,and some lemmas and preparatory knowledge used in the following proof process are given:The expressions of the ordinary least squares estimator(OLSE)and the best linear unbiased estimator(BLUE)under the general form of the general linear model,the restricted linear model and the transformation linear model,the definition and properties of generalized inverse of matrix and some matrix rank formulas are given.The third part of this paper mainly studies the constrained linear model and transformation linear model,some equivalent conditions are given for OLSEMr(K?)=OLSEMt(K?),OLSEMt(K?)=BLUEMr(K?),OLSEMt(K?)=BLUEMt(K?),BLUEMr(K?)=BLUEMt(K?)under restricted models and transformed models to hold.In addition,we also give some results when matrices K take special matrices.The fourth part of this paper mainly studies that equivalent conditions for the equivalence of the ordinary least squares estimator(OLSE)and the best linear unbiased estimator(BLUE)of parametric functions under restricted linear model and some special forms of transformation linear model.The fifth part summarizes the whole paper and looks forward to the future research work.