DOC PREVIEW
ISU STAT 511 - Homework # 1 -2008

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

1Stat 511 HW#1 Spring 2008 1. Write out the following models of elementary/intermediate statistical analysis in the matrix form =+YXβε: a) A special two-variable quadratic polynomial regression model 2011 21 32iiiiiyxxxαααα ε=+ + + + for 1, 2, , 5i = … b) A two-factor ANOVA model without interactions ijk i j ijkyμαβε=+ + + for 1, 2, 1, 2, and 1, 2ij k== =. 2. Find two different generalized inverses of the matrix 110110101⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦ See, for example, panels 162-168 of Koehler’s notes or use the following algorithm (from a book by Searle): To find a generalized inverse for an nk× matrix A of rank r : 1. Find any non-singular rr× sub-matrix of A , C . (It is not necessary that the elements of C occupy adjacent rows or columns of A.) 2. Find ()11 and −−′CC. 3. Replace the elements of C with the elements of ()1−′C . 4. Replace all other elements of A with 0's. 5. Transpose the resulting matrix. Then load the MASS package in R and use the function ginv() provided by that package to find a generalized inverse. Does R return either of the generalized inverses you found “by hand”? 3. Consider the one-way ANOVA model (example b)) from class, version 2. Suppose, however, that there are 4 treatments (groups) and the sample sizes are respectively 2,1,1,3 for treatments 1 through 4. Show that the perpendicular projection matrix onto ()C X (for observations in Y listed in the order of group index) is .5 .5 0 0 0 0 0.5 .5 0 0 0 0 00010 0 0 00001 0 0 00 0 0 0 .33 .33 .330 0 0 0 .33 .33 .330 0 0 0 .33 .33 .33⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦ Do this two ways. First apply the following theorem (Theorem B.33 of Christensen).2Theorem B.33 M is a perpendicular projection matrix onto ()C M if and only if =MM M and ′=MM. (You’ll need to reason that every column of X can be written as a linear combination of columns of M and vice versa, so that () ( )CC=XM.) Then secondly, use the construction for XP (involving a generalized inverse) given in class. 4. In the context of Problem 3, Suppose that ()2,1,4,6,3,5,4′=Y . a) Find ˆY , the least squares estimate of E=YXβ . Show that there is no sensible way to identify an “ordinary least squares estimate of β ” by finding two different vectors b with ˆ=Xb Y. b) Use R and compute all of () ()()ˆˆˆ ˆ ˆˆ ˆ ˆ, , ,,, and ′′′′−−−−YYYYYYYYYY YY YY. 5. Stat 542 and Stat 447 are supposed to cover multivariate distributions, and in particular multivariate normal distributions. Chapters 3 and 4 of Rencher cover that material and there are some summary facts in Appendix 7.1 of the 2004 Stat 511 Course Outline on this material. Review those if you need to do so. Then answer the following (making use of R to do matrix calculations). Suppose that ()3MVN ,YμΣ∼ with ()2,2,0′=μ and 402042223⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠Σ a) What is the marginal distribution of 3y? b) What is the (marginal joint) distribution of 13and yy ? c) What is the conditional distribution of 3y given that 12y=? d) What is the conditional distribution of 3y given that 122 and 1yy==− ? e) What is the conditional distribution of 13and yy , given that 21y=− ? f) What are the correlations 12 13 23, , and ρρρ? g) What is the joint distribution of 123uyy y=−+ and 1231vyy=++? 6. (Koehler) Use the eigen() function in R to compute the eigenvalues and eigenvectors of 31115 1113−⎡⎤⎢⎥=− −⎢⎥⎢⎥−⎣⎦V Then use R to find an “inverse square root” of this matrix. That is, find a symmetric matrix W such 1−=WW V. See slides 87-89 and 91of Koehler's notes or Rencher for help with this. Koehler's Result 1.12 is Rencher’s Theorem 2.12D.37. (Koehler) Consider the matrices 4 4.001 4 4.001 and 4.001 4.002 4.001 4.002001⎡⎤⎡ ⎤==⎢⎥⎢ ⎥⎣⎦⎣ ⎦AB Obviously, these matrices are nearly identical. Use R and compute the determinants and inverses of these matrices. (Note that 113−−≈−AB even though the original two matrices are nearly the same. This shows that small changes in the in the elements of nearly singular matrices can have big effects on some matrix operations.) 8. ′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. 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 τμτμττ μττττ ττ ττ+++ ++ − −−− For those that are estimable, find the 71× row vector ()′′′cXX X− that when multiplied by Y produces the ordinary least squares estimate of ′c β . 9. 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 (see HelpÆManuals(in PDF) after starting 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. 10. 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. 11. Consider the quadratic regression of part a) of Problem 1. Write the hypothesis that the pairs ()()11 21 12 22, and ,xxxxhave the same mean response in the form 0H:=Cβ d


View Full Document

ISU STAT 511 - Homework # 1 -2008

Download Homework # 1 -2008
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Homework # 1 -2008 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Homework # 1 -2008 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?