1Stat 511 HW#2 Spring 2008 1. 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 10 of Homework 1. 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 language. 2. 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 2004 class outline to find () ()ˆˆE and Var−−YY YY. (Use the fact that ()ˆ−=−XYY IPY.) Then write ˆˆ⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟−−⎝⎠⎝⎠XXPYYIPYY and use fact 1 of Appendix 7.1 to argue that every entry of ˆ−YY is uncorrelated with every entry of ˆY. b) Theorem 5.2.A of Rencher or Theorem 1.3.2 of Christensen say that if E =Yμ and Var =Y Σ and A is a symmetric matrix of constants, then ()Etr′′=+YAY AΣμAμ Use this fact and argue carefully that ()()()()2ˆˆEranknσ′−−=−YY YY X23. a) In the context of Problem 3 of HW 1 and the fake data vector used in Problem 4 of HW 1, use R and generalized least squares to find appropriate estimates for 1100010100E and 1001010001⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦Y β in the Aitken models with ()121100000140000000 4 1 0 0 0first diag 1,4,4,1,1,4,4 and then 00 11 0 0 000 0 0 1 1000 0 0 1 4 00000004⎡⎤⎢⎥⎢⎥⎢⎥−⎢⎥==−⎢⎥⎢⎥−⎢⎥−⎢⎥⎢⎥⎣⎦VV (Do the necessary matrix calculations in R.) b) For both of the above 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. 4. a) The basic lm function in R allows one to automatically do weighted least squares, i.e. minimize ()2ˆii iwy y−∑for positive weights iw . For the 1V case of the Aitken model of Problem 3, find the BLUEs of the 4 cell means using lm and an appropriate vector of weights. (Type > help(lm) in R in order to get help with the syntax.) b) The lm.gls() function in the R contributed package MASS allows one to do generalized least squares as described in class. For the 2V case of the Aitken model of Problem 3, find the BLUEs of the 4 cell means using lm.gls. (After loading the MASS package, Type > help(lm.gls) in order to get help with the
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