GS5205Final Key.pdf - STAT GR5205 Final Exam Name UNI...

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STAT GR5205 Final ExamName:UNI:Please write your name and UNIThe Fall GR5205 final is closed notes and closed book. Calculators are allowed. Tablets,phones, computers and other equivalent forms of technology are strictly prohibited. Studentsare not allowed to communicate with anyone with the exception of the TA and the professor.If students violate these guidelines, they will receive a zero on this exam and potentiallyface more severe consequences. Students must include all relevant work in the handwrittenproblems to receive full credit.Theory ComponentProblem 1 [10 pts]Part I (5 pts)LetXbe a full ranknpdesign matrix and define the hat matrix asH=X(XTX)-1XT.Recall that the column space ofX, denotedC(X), is the set of all linear combinations ofthe columns inX. Prove that ifv2C(X), thenHv=v.
Part II (5 pts)Letˆβbe the least squares estimator ofβ. Use the result from Problem 1.I to prove that
Problem 2 [25 pts]Consider three models:(1)Yi=β1xi1+i,i= 1, . . . , n,iiidN(0,σ2),(2)Yi=β2xi2+i,i= 1, . . . , n,iiidN(0,σ2),(3)Yi=β1xi1+β2xi2+i,i= 1, . . . , n,iiidN(0,σ2).Denote the respective data vectors and full design matrix byY=(Y1Y2· · ·Yn)=
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Part I (10 pts)Assuming that the vectorsx1andx2are perfectly uncorrelated (orthogonal), prove thatHX= (H1+H2)X
Noteifstudentdirectlyprovedtherelation#=II.+112,thatisgreatalso!PryorNote11
Part II (10 pts)Similarly, assuming that the vectorsx1andx2are perfectly uncorrelated (orthogonal), provethatH(x1+x2) = (H1+H2)(x1+x2)Part III (5 pts)Note:the following exercise is not a proof.Assuming that the vectorsx1andx2areperfectly uncorrelated, state the (obvious) relationship betweenH1,H2, andH.4ProfNotethatx.itn,GGCI)Thutllx,tx,)=IIX,+IIs=I,tIs(II,tHas.lk,-i×,)=H,I,+It,×z+11,4++5×-2=I,+o+OtIsThus-tlX,tx,)=(II,+µ,)(Iit,)µ=th+H-2

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