1 A Primer on Linear Models. 2 Chapter 1 corrections. 3 Chapter 2 corrections. 4 Chapter 3 corrections. 1.1 Corrections 23 May 2015

Transcription

1 A Prmer on Lnear Models. Correctons 23 May Chapter correctons Fx page: 9 lne -7 denomnator ( ρ 2 ) s mssng, should read Cov(e t, e s ) = V ar(a t )/( ρ 2 ) Fx page: lne -5 sgn needs changng: cos(a b) = cos(a)cos(b) + sn(a)sn(b) 3 Chapter 2 correctons Fx page: 5 lne - mssng dot n subscrpt, end of Example 2.2 should read any value of c, where y. = y j /n. Fx page: 23 lne 7 matrx n dsplay s P X, not I P X ; should read (I P X )z = = Fx page: 27 lne 4 error e should be ê for resduals orthogonalty of the resduals ê to the columns of the desgn matrx X Fx page: 3 lne dmenson of matrx A should be p p Fx page: 34 lne 7 mssng sgn choosng u = x + se () leads to Ux = se () 4 Chapter 3 correctons Fx page: 44 lne 4 parameter vector b s mssng, should read N n n 2... n a n n 0 0 X T Xb = n 2 0 n n a n a µ α α 2... α a = Ny.. n y. n 2 y n a y a. = XT y

2 Fx page: 46 lne 2 3, 2 element of X T X matrx should be b T a Fx page: 46 lne - n should be b seven of the a b = 4 3 = 2 cells are observed. Fx page: 49 lne 5 frst vector n second dsplay should have frst component equal to 2, not, so the vector s ( 2,,,,,, 0, 0, 0, 0, 0, 0) 5 Chapter 4 correctons Fx page: 75 lne 2 a should be d where d T X = λ T. Fx page: 75 lne 3 lnear should be lnearly Fx page: 83 lne -, -7, and -0 mssng transposes and... = (y Xb) T V (y Xb) = (y Xˆb GLS ) T V (y Xˆb GLS ) ˆσ 2 GLS =... = (y Xˆb GLS ) T V (y Xˆb GLS )/(N r) Fx page: 84 lne 7 mssng subscrpt of σ 2 Then t can be shown that V ar(e ) = σa/( 2 ρ 2 ) and the covarance Fx page: 86 lne 6 mssng σ 2 σ 2 VX = (σ 2 I N + τ 2 N T N ) Fx page: 88 lne 3,4 subscrpt errors E(y. ) = β 0 + β + β 2 n x 2 j j = β 0 + β + β 2 x 2. + β 2 n (x 2 j x. ) 2 Fx page: 9 lne -5 Example 4.7 should be Example 4.8 Fx page: 92 lne -5 transposes n wrong place, should read... f R s square and nonsngular and RVR T = I, then... Fx page: 95 lne -7 square n wrong place n Exercse 4.28, should read j 2

3 V ar( ˆβ ) = σ 2 /( (x j x.. ) 2 ) j Fx page: 96 lne 7,8 mssng parentheses n second pece, should read V ar((µ + e) T P(µ + e)) = E (µ + e) T P(µ + e)(µ + e) T P(µ + e) (E (µ + e) T P(µ + e) ) 2 Fx page: 07 lne both fgures the horzontal axes (u) n both graphs should go to 30, not 3 6 Chapter 5 correctons Fx page: 5 lne 4 In ANOVA table for Example 5.3, the noncentralty parameter for group s mssng n n the sum and should read a n(α α)2 = Fx page: 7 lne 5 projecton matrx n denomnator of (5.) s ncorrect, should read r 2 = yt (P X P )y y T (I P )y. Fx page: 7 lne -8 rght hand sde of (5.2) should be squared... and the expresson s the squared sample correlaton between the response and ftted values R 2 = ( (ŷ y)(y y)) 2 (ŷ y) 2 (y y) 2 Fx page: 23 lne ndex subscrpt s upper case Let X N, N =, 2,... be a sequence of random... 7 Chapter 6 correctons Fx page: 25 lne -7 Corollary 5.2, not (nonexstent) Corollary 5.5 Fx page: 26 lne -2,-3 no need for gant braces; equaton (6.4) should read f(y b, σ 2 ) = (2π) N/2 (σ 2 ) N/2 exp{ 2σ 2 (bt X T Xb)} exp{w (b, σ 2 )T (y)+w 2 (b, sgma 2 )T 2 (y)}. 3

4 Fx page: 27 lne - change sgn of second term (N/2) σ 2 + 2(σ 2 ) 2 Q(ˆb) Fx page: 33 lne 7 mssng n n frst sum a a = n (y. y. ) 2 n (y N. y. ) =2 Fx page: 42 lne -9 and -9 τ should be m n the probablty statements or =2 P r( t j < t N r,α/2 for all j) = P r(m B) < α P r( t j t N r,α/2 for at least one j) = P r(m / B) > α Fx page: 46 lne -6 constant and degrees of freedom should both be a(n )... and ndependently a(n )ˆσ 2 /σ 2 χ 2 a(n ) Fx page: 46 lne -3 degrees of freedom n Equaton (6.29) should be a(n ) to read 2 (y. y j. ) ˆσ n q a,a(n ) α α j (y. y j. ) + ˆσ n q a,a(n ) Fx page: 47 lne 6,7 degrees of freedom n Equaton (6.30) should be a(n ) to read u y. ˆσ q n a,a(n ) u u τ u y 2. + ˆσ q n a,a(n ) 2 u Fx page: 5 lne -4 Prove Corollary 6., not 6.2. Fx page: 5 lne -2 add where to evaluate dervatve normal Gauss-Markov model wth respect to σ 2 evaluated at σ 2 = ˆσ MLE 2 s negatve: Fx page: 52 lne -8,-7 reword second sentence of Exercse 6.9 Construct the remanng a + n lnearly ndependent functons of γ j s and show how they are confounded wth the man effects α s and β j s. Fx page: 53 lne 3 Add nstructon to Exercse 6. Use σ 2 = 0.,, 0. Fx page: 53 lne 9 Correct the Cauchy-Schwarz nequalty (u T w) 2 u 2 w 2 4

5 8 Chapter 7 correctons Fx page: 73 lne -4 mssng parentheses, wrong projecton matrx n denomnator F = yt (P Z P X )y/(r(z) r(x)) y T. (I P Z )y/(n r(z)) 9 Chapter 8 correctons Fx page: 8 lne - drop bar to fx expresson for SSE SSE = y T (I P X )y = Fx page: 83 lne 3,4 dvde by upper case N, not lower case n 2φ a σ 2 /N = N a n (y j y. ) 2 = j= n α 2 α 2 = α T N Dα N 2 αt D T Dα = α T Fx page: 83 lne 9 long expresson n exponent n equaton (8.4) E(E(e Us α s)) = ( 2sσ 2 ) (a )/2 E(e sσ2 /( 2sσ 2 ) 2λ a ) Fx page: 88 lne -8 γ not tau by wrtng γ j = v j v. Fx page: 88 lne -6 dsplay not broken up N D N 2 DT D α γ. = v.. γ.j γ.. = v.j v.. γ j γ. γ.j + γ.. = v j v. v.j + v.. Fx page: 89 lne 20 subscrpt upper case N e N N (0, R) Fx page: 92 lne -7 Result A.8(e), not A.7(e) Fx page: 93 lne 4 determnant as wrtten s not correct, should read X T V X = a = n /(σ 2 + n σ 2 a). 5

6 Fx page: 200 lne 5 correct mean y E y X = Fx page: 200 lne 23 subscrpt, not superscrpt * whose varance s σ 2 (a T a + ) = σ 2 (x T (X T X) g x + ). Fx page: 200 lne 26 subscrpt, not superscrpt * Fx page: 200 lne 27 subscrpt, not superscrpt * P r( t α/2 < Fx page: 200 lne 28,29 subscrpt, not superscrpt * x T b x T ˆb y N(0, σ 2 (x T (X T X) g x + )) x T ˆb y ˆσ(x T (X T X) g x + ) /2 < t α/2) = α = P r(x T ˆb t α/2ˆσ(x T (X T X) g x +) /2 < y < x T ˆb+t α/2ˆσ(x T (X T X) g x +) /2 ) Fx page: 200 lne 3 or -3 subscrpt, not superscrpt * Fx page: 20 lne sgn of second term x T ˆb ± t α/2ˆσ(x T (X T X) g x + ) /2. V ar(a T y y ) = a T Ωa 2a T ω + ω. Fx page: 20 lne 3 mssng zero RHS Unbasedness agan means that a T Xb x T b = 0 for all b, or X T a x = 0. Fx page: 20 lne 5 change both sgns Fx page: 204 lne -0 Exercse 6.7 should be Exercse 6.25 Ωa ω Xλ = 0, 0 Chapter 9 correctons Fx page: 20 lne -0 second (I N P X ) s correct, but superfluous and msleadng... = tr (I N P X )(XB.k B Ṭ jx T + Σ jk I N ) = (N rank(x))σ jk. 6

7 Appendx A correctons Fx page: 250 lne 5 subscrpt of dentty matrx should be n r nstead of m r, should read x x 2 = B b B B 2x 2 x 2 = B b 0 + Fx page: 263 lne -6 subscrpt mssng on matrx on left hand sde of dsplay, should read Fx page: 264 lne -5 mssng transpose, should read G 2 = C Ir E E 2 E 2 E A T (AA T ) = A + B Fx page: 267 lne -8 frstmatrx s not postve defnte, so change the (3, 3) element to Fx page: 268 lne -4 where to evaluate the dervatve... wth respect to t at t = 0 s equal to... Fx page: 268 lne - ncorrect formula for Bnomal Inverse Theorem; should read B B 2 I n r (A + UBV) = A A U(B + VA U) VA 2 Acknowledgements x 2 Great thanks are due Profs. Jerry M. Davs, Mohsen Pourahmad, Josh Tebbs, Jm Robson-Cox, Len Stefansk, Howard Bondell, and Guowe L for ther contrbutons to ths lst. JF Monahan, last update 23 May 205 7