A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications

Abstract Suppose that a weakly singular linear regression model <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">M</m:mi> </m:math> {\mathscr{M}} and its two competing restricted models <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {{\mathscr{M}}}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\mathscr{M}}}_{2} are given. We first compare the superiorities of the best linear unbiased estimators (BLUEs) under <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">M</m:mi> </m:math> {\mathscr{M}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {{\mathscr{M}}}_{1} related to the covariance matrix criterion and derive some necessary and sufficient conditions. Based on these conditions and the covariance matrix criterion, we then compare the superiorities of the BLUEs under <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {{\mathscr{M}}}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="script">M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\mathscr{M}}}_{2} and obtain some results. We finally apply these results to measure the influence of an observation on the BLUE.