A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models

Consider the partitioned linear regression model A=(y,Xβ+Xβ,σV)and its fourreduced linear models,where y is an n×1 observable random vector with E(y)=Xβ and dispersionmatrix Var(y)=σ~2V,where σ~2 is an unknown positive scalar,V is an n×n known symmetric non-negative definite matrix,X=(X:X)is an n×(p+q)known design matrix with rank(X)=r(?)(p+q),and β=(β′1:β′)with βand βbeing p×1 and q×1 vectors of unknown parameters,respectively.Inthis article the formulae for the differences between the best linear unbiased estimators of MXβunderthe model A and its best linear unbiased estimators under the reduced linear models of A are given,where M=I-XX~+.Furthermore,the necessary and sufficient conditions for the equalities betweenthe best linear unbiased estimators of MXβunder the model A and those under its reduced linearmodels are established.Lastly,we also study the connections between the model A and its lineartransformation model.