Research On Equivalence Of Predictor/Estimator Under Linear Models

Linear statistics model is the most basic model in regression analysis.In statistics,it has been playing a very important role from theoretical and applied points of view,which are considered to be very important part of current statistics theory,many phenomena in biology,medicine,economics,management,geology,meteorology,agriculture,industry,engineering and technology can be approximately described by linear models,and become one of the most widely used models in modern statistics.Linear model is a general term for a class of statistical models,including linear regression model,variance analysis model,covariance analysis model,mixed effects model(or variance component model),longitudinal data model and growth curve model,it is a kind of statistical model only solid theoretical support in linear algebra and matrix theoryThis paper mainly reexamines equivalence between Ordinary Least Squares Estimator(abbreviated as OLSE)and Best Linear Unbiased Estimator(abbreviated as BLUE)of mean parameter vector and partial mean parameter vector under general linear model with parameter constraints(CGLM)from view of algebra by using various algebraic analysis tools in linear algebra and matrix theory,relationships between Best Linear Unbiased Predictors(abbreviated as BLUPs)of all unknown parametric function under general linear model and its mis-specified form,and equivalence between BLUP/BLUE of all same parametric function under multivariate general linear model(MGLM)and multivariate linear model with over-parameterized(over-fitted),is one of the classical research problems in theory of regression analysis.The paper is divided into eight parts.Introduction summarizes the research background,significance,related literature,research contents,methods and innovation pointsIn the first chapter,we mainly introduce general form of two partitioned and multiple partitioned linear model without parameter constraints,with parameter constraints and multivariate general linear model,definitions of estimability of unknown parametric function and unknown parameter matrix function,ordinary least squares estimator(OLSE)and best linear unbiased estimator(BLUE),and the analytical expressions of OLSE and BLUEThe second chapter mainly gives all kinds of necessary and sufficient conditions for matrix rank and spatial relation formula of unknown parametric function,mean parameter vector and partial mean parameter vector estimability under two block linear model with partial parameter restraints and analytic expressions of OLSE and BLUE,which contain known matrix in the model and complex matrix operator of their generalized inverse.The algebraic and statistical properties of estimators are given by using various analytical tools in matrix theory.The relationships between OLSEs and BLUEs of mean parameter vector and partial mean parameter vector in two block linear models with partial parameter restraints are investigated.We first review and give some known and novel equivalent conditions on equivalence of OLSE and BLUE under general linear model,then give novel equivalent results of OLSEs and BLUEs of mean parameter vector and partial mean parameter vector in two block linear model with partial parameter restraints.At last,an important conclusion is given for relationship between mean parameter and partial mean parameter vector under two block linear models with partial parameter restraints.That is,the following statements are equivalent:(a)OLSEpr(X1?1+ X2?2)-BL UEpr(X1?1 +X2?2)holds definitely.(b)OLSEpr(X1?1 = BL UEpr(X1?1)and OLSEpr(X2?2)= BLUEpr(X2?2)hold definitely.(c)OLSEpr(X2X1,?1,)= BL UEpr(X2-X1,?1)and OLSEpr(X1X2?2)= BL UEpr(X1X2,?2)hold definitely.In the third chapter,we give a new necessary and sufficient conditions for estimability of unknown parametric function under general linear model and two block linear model with parameter constraints,analytic expressions for OLSE and BLUE of unknown parametric function,examine relationships between OLSEs and BLUEs of mean parameter vector and partial mean parameter vector,give a variety of novel equivalent conclusions of OLSE and BLUE under linear model with parameter constraints.Finally,the important conclusion of the relationship between mean parameter and partial mean parameter vector is given,that is,the following four statistical facts are equivalent:(a)OLSE?(X1?1+X2?2)-BLUE?(X1?1 +X2?2)holds definitely(with probability 1).(b)OLSE?(X1,?1+X2?2)-BLUE?(X1,?1+X2,?2)holds definitely(with probability 1).(c)OLSE?(X1?1)= BLUE?(X1?1)and OLSE?(X2?2)=BLUE?(X2?2)hold definitely(with probability 1).(d)OLSE?(X1?1)= BL UE?(X1?1)and OLSE?(X2?2)= BL UE?(X2?2)hold definitely(with probability 1)These results show that many statistical facts that are analyzed and deduced in linear model are actually equivalent,in other words,there is a variety of statistical explanations on equivalence of OLSE and BLUE under linear model In the fourth chapter,we study the relationships between OLSEs and BLUEs of mean parameter vector and partial mean parameter vector under multiple block linear model with parameter constraints,give a variety of novel algebraic and statistical explanations on equivalence of OLSE and BLUE by using a variety of classical and new algebraic tools and methods.These results show essential link between OLSE and BLUE under various general assumptions,so that we can make full use of these equivalent facts in different situations.It can be served as general reference for statistical inference under linear model with parameter constraints.In the fifth chapter,it is studied that a real linear regression model may be represented in mis-specified form for some reason in statistical analysis.Predictor of all unknown parametric function under mis-specified model will lead to wrong conclusion in statistical inference of real linear model.Based on the exact algebraic tools in matrix theory,we study relationships between best linear unbiased predictor(BLUP)of all unknown parametric function under real linear model and its mis-specified forml,give equivalent conditions of all kinds of relationships between BLUPs under two models.That is,the following statements are equivalent:(a)there exist Pk0;J0;X0;?0 and PK;J;X;? such that PK0;J0;X0;?0=PK;J;X;?·(b){BLUP?(?)}?{BLUP?0(?0)}?(?)holds definitely(with probability 1)(c){BLUP?(?)}?{BLUP?0(?0)}?(?)holds with probability 1(d){BLUP?(?)}(?){BLUP?0(?0)} holds with probability 1(e)R(M2)(?)R(N2)Multivariate general linear model(MGLM)is relative direct extensions of univariate general linear model(UGLM),which normally means the incorporation of regressing one response variable on a given set of regressors to several response variables on the regressors.In statistical inference,we often need to analyze and compare if there is equivalence or similarity between two or more competing statistical models.We often face with the task of comparing two or more linear regression models by adding or deleting regressors once the observed matrix of the response variables are given.If taking original model as a true model,over-parameterized(over-fitted)version is obtained by adding new component part incorrectly.Predictors/estimators dernved from the two models are not necessarily the same.In the sixth chapter,we first establish exact closed-form formulas for BLUP/BLUE of all common parameter matrices forming functions and its special cases under the two models by using closed-form solutions of constrained quadratic matrix-valued function optimization problem in the Lowner partial ordering,then give valuable algebraic and statistical properties of BLUP/BLUE and decomposition equality of BLUP.Finally,we give all kinds of novel equivalent conditions for BLUP/BLUE equal definitely(with probability 1,variance)all common parameter matrix function and its special cases under two models.The seventh chapter mainly introduces the follow-up work of matrix analysis tools for additive decomposition and equivalence on predictor/estimator under random effects model and multivariate linear models.The innovation point as follow:Equivalence between basic estimators,ordinary least squares estimator(OLSE)and best linear unbiased estimator(BLUE)and equivalence between best linear unbiased predictors(BLUPs)under different models is one of the classical research problems in regression analysis theory.Rank of matrix is one of the most basic concepts and numerical features in linear algebra and matrnx theory.It is the most significant finite non-negative integer in reflecting intrinsic properties of matrices.In matrix theory and application,a large number of analytical formulas for matrix rank have been established,matrix rank method has been used to obtain fruitful research results in statistical inference of linear model.Although there are many classic inferences about linear statistical model,it is also possible to make full use of various effective matrix analysis tools to find out all kinds of novel and insightful conclusions,give exact analytical expressions,a large number of algebraic and statistical properties of these basic predictors/estimators under most general assumptions for model matrix,and additive decomposition of BLUP,a lot of novel and easy to use equivalent conditions and statistical explanations for OLSE and BLUE.Furthermore,these excellent properties can be better applied to statistical inference of linear regression models.This work belongs to innovation research of basic theory of regression analysis,and uses a series of novel and effective analytical tools and methods in mathematics.It has enriched and developed core content of regression analysis,and has a profound and long-term influence on research of statistical inferenceSpecifically,there are the following features:1.Under most general assumptions for model matrix and covariance matrix,according to definitions of two predictors/estimators equal definitely(with probability 1),equivalence between OLSE and BLUE is reexamined by algebraic analytical tools in linear algebra and matrix theory from algebraic point of view,give equivalent conditions of matrix rank and spatial relationship on estimability for unknown parametric function,mean parameter vector and partial mean parameter vector under two block linear model with partial parameter restraints,analytic expressions of OLSE and BLUE and all kinds of novel equivalent conditions between them.At last,an important conclusion is given for relationship between mean parameter and partial mean parameter under two block linear model with partial parameter restraints.So this is a very worthwhile question.2.Under most general assumptions for model matrix and covariance matrix,according to definitions of two predictors/estimators equal definitely(with probability I),equivalence between OLSE and BLUE is reexamined by algebraic analytical tools in linear algebra and matrix theory from algebraic point of view,give equivalent conditions of matrix rank and spatial relationship on estimability for unknown parametric function,mean parameter vector and partial mean parameter vector under two block and multiple block linear model with parameter constraints,analytic expressions of OLSE and BLUE and a lot of novel and unprecedented equivalent conditions between them.In other words,there are many different statistical explanations for the equivalence of OLSE and BLUE under linear model.3.Under most general assumptions for model matrix and covariance matrix,give exact analytical espressions of BLUP of all known parametric function,a large number of algebraic properties,decomposition equality of BLUP and novel equivalent conditions of matrix rank and spatial relationship under real model and its mis-specified form by algebraic analytical tools in linear algebra and matrix theory from algebraic point of view.4.Under most general assumptions for model matrix and covariance matrix,give exact analytic expressions of BLUP/BLUE of all common parameter matrix function and its special cases,large number of algebraic and statistical properties decomposition equality of BLUP and novel equivalent conditions of matrix rank and spatial relationship under multivariate general linear model(MGLM)and its over-parameterized(over-fitted)form by algebraic analytical tools in linear algebra and matrix theory from algebraic point of view.