Download Linear Regression Analysis, Second Edition by George A. F. Seber, Alan J. Lee(auth.) PDF

By George A. F. Seber, Alan J. Lee(auth.)

Concise, mathematically transparent, and complete therapy of the subject.
* elevated assurance of diagnostics and techniques of version fitting.
* calls for no really expert wisdom past an exceptional seize of matrix algebra and a few acquaintance with straight-line regression and easy research of variance models.
* greater than 2 hundred difficulties through the publication plus define ideas for the exercises.
* This revision has been generally class-tested.Content:
Chapter 1 Vectors of Random Variables (pages 1–16):
Chapter 2 Multivariate general Distribution (pages 17–33):
Chapter three Linear Regression: Estimation and Distribution concept (pages 35–95):
Chapter four speculation trying out (pages 97–118):
Chapter five self assurance durations and areas (pages 119–137):
Chapter 6 Straight?Line Regression (pages 139–163):
Chapter 7 Polynomial Regression (pages 165–185):
Chapter eight research of Variance (pages 187–226):
Chapter nine Departures from Underlying Assumptions (pages 227–263):
Chapter 10 Departures from Assumptions: analysis and treatments (pages 265–328):
Chapter eleven Computational Algorithms for becoming a Regression (pages 329–389):
Chapter 12 Prediction and version choice (pages 391–456):

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Extra resources for Linear Regression Analysis, Second Edition

Example text

Of Z is £[exp(t'Z)] = t=i II ex P(H) i=l exp(U't). 4) Now if Y ~ JV n (/z,£), we can write Y = E X / 2 Z + /x, where Z ~ N n ( 0 , I n ) . 4) and putting s = 531/2t, we get JS[exp(t'Y)] = = E [ e x p { t ' ( S 1 / 2 Z + /*)}] £[exp(s'Z)]exp(t'/x) = exp(|s's)exp(t'/x) = = e x p ^ t ' E ^ E ^ t + tV) exp(t'/x + | t ' E t ) . 5) Another well-known result for the univariate normal is that if Y ~ N(n, a2), then aY + b is N(afi + b,a2cr2) provided that o ^ O . A similar result is true for the multivariate normal, as we see below.

MISCELLANEOUS EXERCISES 1 1. rY{E[X\Y]}. Generalize this result to vectors X and Y of random variables. 2. Let X = (X1,X2,X3)' with Var[X] = / 5 2 3 \ 2 3 0 . \ 3 0 3 / (a) Find the variance of Xi — 2X2 + X3. (b) Find the variance matrix of Y = (Yi,^)', where Y\ = X\ + X2 and Y2 = Xi + X2 + X3. 3. Let X\, X2, ■ ■ ■ ,Xn be random variables with a common mean \i. Suppose that covfA'j, Xj] = 0 for all i and j such that j > i + 1. If Q1 = JT(Xi-X)2 t=l and Q2 = {Xx - X2f + (X2 -X3)2 + --- + (AVi - Xn)2 + (Xn - X,)2, 16 VECTORS OF RANDOM VARIABLES prove that E n(n - 3) = var[A-].

7, will be x£ if and only if R ' A R is idempotent of rank r. However, this is not a very useful condition. A better one is contained in our next theorem. 8 Suppose that Y ~ iV n (0,S), and A is symmetric. Then Y ' A Y is x\ */ and onh */ r °f the eigenvalues of A S are 1 and the rest are zero. Proof. 2). 2), r = rank(R'AR) = t r ( R ' A R ) = t r ( A R R ' ) = tr(AS). 1), R ' A R and A R R ' = A S have the same eigenvalues, with possibly different multiplicities. Hence the eigenvalues of A S are 1 or zero.

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