Proving Linear Estimator (beta) is BLUE?

In the book Statistical Inference pg 570 of pdf, There’s a derivation on how a linear estimator can be proven to be BLUE.

I got all the way up to 11.3.18 and then the next part stuck me.

After finding the dis that satisfy the condition: Σdi=0 & Σdixi=1, we take 11.31.7 derived from the lemma:
ni=1dixi=ni=1K(xiˉx)xi=KSxx

Where did they get Sxx from? Isn’t that supposed to be defined as ni=1(xiˉx)2 which does not equal to the equation above.

Furthermore, if I’m on the right tracking then I’m not sure how 11.3.19 is worked out.

I’m so close to figuring it out guys and I would appreciate some guidance.

Answer

Sxx=ni=1(xiˉx)2=ni=1(x2i2xiˉx+ˉx2)
=(ni=1x2i)nˉx2=(ni=1x2i)ˉxni=1xi=ni=1x2iˉxxi=ni=1(xiˉx)xi

Attribution
Source : Link , Question Author : Kevin Pei , Answer Author : TenaliRaman

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