Stat 511 HW#2 Spring 2004 1. ′c β is estimable exactly when ()C′∈cX. This occurs exactly when ′=XcPc, that is when c is its own projection onto ()C′X . Clearly, ()−′′′=XPXXXX. Use R and find this matrix for the situation of Problem 3 of HW 1. Then use this matrix and R to decide which of the following linear combinations of parameters are estimable in this example: ()()1112 1212 1234, , , 2 , , and τµτµτ τ µτ ττ τ τ τ τ τ+++ ++ − −−− For those that are estimable, find the 61× row vector ()′′′cXX X− that when multiplied by Y produces the ordinary least squares estimate of ′c β . 2. Twice now you've been asked to compute projection matrices in R. It seems like it would be helpful to “automate” this. Have a look at Chapter 10 of An Introduction to R. Write a function (call it, say, project) that for an input matrix produces a matrix that projects vectors onto the column space of the input matrix. Test your function by running it on both and ′XX for the situation of Problem 3 of HW 1. 3. Consider the (non-full-rank) “effects model” for the 22× factorial (with 2 observations per cell) called example d in lecture. a) Determine which of the parametric functions below are estimable. ()11 2 1 1 1 1 11 11 12 11 22 21,, , ,,αα α µ α β µ α β αβ αβ αβ αβ αβ αβ− ++ +++ − − − For those that are estimable, find the 81× row vector ()′′′cXX X− that when multiplied by Y produces the ordinary least squares estimate of ′c β . b) For the parameter vector β written in the order used in class, consider the hypothesis 0H: =Cβ 0 , for 01 1000 0 0 0000001 1 11−=−−C Is this hypothesis “testable”? Explain. 4. Consider the one-variable quadratic regression of Problem 1 of HW1. Write the hypothesis that the values and xx′ have the same mean response in the form 0H: =Cβ d . 5. Suppose we are operating under the (common Gauss-Markov) assumptions that E =ε 0 and 2Varσ=ε I . a) Use fact 1. of Appendix 7.1 of the class outline to find () ()ˆˆE and Var−−YY YY. (Use the fact that ()ˆ−=−XYY IPY.) b) Then write ˆˆ=−−XXPYYIPYYand use fact 1 of Appendix 7.1 to argue that every entry of ˆ−YY is uncorrelated with every entry of ˆY . c) Use Theorem 5.2.A, page 95 of Rencher to argue that ()()()()2ˆˆEranknσ′−−=−YY YY X 6. In the context of Problem 3 of HW 1 and the fake data vector used in Problem 4 of HW 1, use R and weighted generalized least squares to find appropriate estimates for 1100010100E and 1001010001Y β in the Aitken models with ()1211000014000000 4 1 0 0first diag 1, 4, 4,1,1, 4 and then 00 1 1 0 000 0 0 1 100 0 0 1 4−==−−−VV For both of these covariance structures, compare the (Aitken model) covariance matrices for generalized least squares estimators to (Aitken model) covariance matrices for the OLS estimators of EY and the Cβ above. 7. In class Vardeman argued that hypotheses of the form 0H:=Cβ 0 can be written as ()00H:E C∈YX for 0X a suitable matrix (and ()()0CC⊂XX). Let’s investigate this notion in the context of Problem 3. Consider 000001 1 1 101 100.5.5 .5 .5−−=−−−C and the hypothesis 0H: =Cβ 0 . a) Find a matrix A such that =CAX. b) Let 0X be the matrix consisting of the 1st, 4th and 5th columns of X . Argue that the hypothesis under consideration is equivalent to the hypothesis ()00H:E C∈YX. (Note: One clearly has ()()0CC⊂XX. To show that () ()0CC⊥′⊂XA it suffices to show that 0′=APX 0 and you can use R to do this. Then the dimension of ()0C X is clearly 2, i.e. ()0rank 2=X . So ()0C X is a subspace of () ( )CC⊥′XA∩ of dimension 2. But the dimension of () ( )CC⊥′XA∩ is itself ()()rank rank 4 2 2−=−=XC .) c) Interpret the null hypothesis under discussion here in Stat 500
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