1 Stat 511 Exam1 February 24, 2009 Prof. Vardeman I have neither given nor received unauthorized assistance on this exam. ________________________________________________________ Name _______________________________________________________________ Name Printed21. Consider a segmented simple linear regression problem in one variable, x. In particular, suppose that 6n = values of a response y are related to values 0,1,2,3,4,5x= by a Gauss-Markov normal linear model =+YXβε for 112203314425566100110120, , , and 131142153yyyyyyεεβεβεβεε⎛⎞ ⎛⎞⎛⎞⎜⎟ ⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎛⎞⎜⎟ ⎜⎟⎜⎟⎜⎟== = =⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟⎝⎠⎜⎟ ⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ ⎜⎟⎝⎠⎝⎠ ⎝⎠YX βε Values of x are in the second column of the model matrix. This model allows the linear form 01yxββ≈+ for 2x ≤ and the linear form ()()011222yxββββ≈+++ − for 2x ≥ . Notice that there is continuity of these forms at 2x=. a) This a full rank model. Argue carefully that this is the case. Here ()1.825 .474 .526.474 .421 .579.526 .579 .921−−⎛⎞⎜⎟′=− −⎜⎟⎜⎟−⎝⎠XX and for ()0,2,4,3,1,0′=Y , ()1.0182.0533.447−−⎛⎞⎜⎟′′=⎜⎟⎜⎟−⎝⎠XX XY and .202SSE = . b) Is there definitive evidence that a simpler model 01 yxxββ≈+∀ is inadequate here? Explain. 10 pts 10 pts3c) Tomorrow a total of 3 new observations are to be drawn from this model at, respectively, 1, 2, and 3x = . Call these ** *12 3, , and yy y. The quantity ()()** ** * **32 21 3 212yyyyyyy−−−=−+ is an empirical measure of change in slope of mean y as a function of x at 2x= based on these new observations. Provide 95% two-sided prediction limits for this quantity. (Plug in completely, but you need not do arithmetic.) d) Find the value and degrees of freedom for a t statistic for testing 0|1 |5H:yx yxμμ=== (the hypothesis that the mean responses are the same for 1 and 5xx== ). T =_____________ df=___________________ 10 pts 10 pts4e) Write out (plug in completely so that your implied answer is numerical, but you need not do the arithmetic) a test statistic that you could use to test the hypothesis that 0|1 |5H: 1yx yxμμ====. Say exactly what null distribution you would use. f) It is possible to compute ()1−′′XXX X both for the full model specified at the beginning of this problem and for a model with X matrix consisting of only the first two columns of the original one. The diagonal entries of these two matrices are in the table below. i 1 2 3 4 5 6 ix 0 1 2 3 4 5 diagonal entry of ()1−′′XXX X for the original X.825 .298 .614 .272 .298 .693diagonal entry of ()1−′′XXX X for the reduced X.524 .295 .181 .181 .295 .524 Compare the two patterns above and say why (in the context provided at the beginning of this problem) they "make sense." 10 pts 10 pts52. Suppose that Y is ()2MVN ,nσμI and that , , and AB C are symmetric nn× matrices with =AB 0 , =AC 0 , and =BC 0 . Argue carefully that the three random variables ′YAY,′YBY, and ′YCY are jointly independent. 3. Suppose that 11 12 and yy are independent ()1N,μηvariables independent of 21 22 and yy that are independent ()2N,4μη variables. (The and 4ηη are variances.) What is the BLUE of 12μμ−? Explain carefully. 10 pts 10 pts64. a) For any non-zero n∈ℜw the set of multiples of w , namely {}|cc∈ℜw , is a 1-dimensional subspace of nℜ . We might call this subspace ()C w. Consider the operation of perpendicular projection onto ()C w , accomplished using the nn× projection matrix wP . Argue carefully that for any n∈ℜv , ′⎛⎞=⎜⎟′⎝⎠wvwPv www (Note that c=wPv w for some c∈ℜ, and consider c′ww.) b) In the regression context from lecture, let ()12 1|||| |rr−=X1xx x x" and ()112 1||||rr−−=X1xxx" . Further, let ()11rrrr r r−−=− =−XXzxPx IP x Argue carefully that for any ()1rC−∈vX, r⊥vz. 5pts 5pts7c) As a matter of fact, 1rr−−=XX zPP P. Argue carefully here that 1r−−XXPP is symmetric and idempotent, and that ()1r−−=XXPP vv for any ()rC∈vz. d) Using the facts in a)-c) argue carefully that 11ˆrrrrrr−−⎛⎞′=+⎜⎟′⎝⎠XezYPY zzz for ()11rr−−=−XeIPY. Then say why it is clear that the multiplier of rz here is OLSrb , the ordinary least squares estimate of the regression coefficient rβ in the full original regression. What interpretation does this development provide for OLSrb ? 5pts