Fundamentals of Probability and Statistics for Engineers

Linear Models and Linear Regression 339at ^ and ^ . Elementary calculations show thatandq 2 Q ˆ 2n > 0;q^2D ˆ 4n Xniˆ1…x i x† 2 > 0The proof of this theorem is thus complete. Note that D would be zero if allx i take the same value. Hence, at least two distinct x i values are needed for thedetermination of ^ and ^.It is instructive at this point to restate the foregoing results by using a morecompact vector–matrix notation. As we will see, results in vector–matrix formfacilitate calculations. Also, they permit easy generalizations when we considermore general regression models.In terms of observed sample values (x 1 ,y 1 ), (x 2 ,y 2 ),...,(x n ,y n ), we have asystem of observed regression equationsy i ˆ ‡ x i ‡ e i ; i ˆ 1; 2; ...; n: …11:11†Letand let2 3 2 3 2 31 x 1 y 1 e 11 x 2C ˆ 6 . . 74 . . 5 ; y ˆ y 26 . 745 ; e ˆ e 26 . 74 . 5 ;1 x ne nq ˆy n:Equations (11.11) can be represented by the vector–matrix equationy ˆ Cq ‡ e:…11:12†The sum of squared residuals given by Equation (11.6) is nowQ ˆ e T e ˆ…y Cq† T …y Cq†: …11:13†TLFeBOOK