Stts 443.3 & 85.3 Summr
The Woodbur Theorem BCD B C D B D where the verses C C D B, d est.
Block Mtrces Let the m mtr m q q m be rttoed to sub-mtrces,,,, Smlrl rtto the m k mtr B B B mk m B B l kl
Product of Blocked Mtrces The B B B B B B B B B B B B B
The Iverse of Blocked Mtrces Let the mtr be rttoed to sub-mtrces,,,, Smlrl rtto the mtr Suose tht B = - B B B B B
Summrzg Let Suose tht - = B the B B B B B B B B B B
Smmetrc Mtrces mtr,, s sd to be smmetrc f Note: B B B B
The trce d the determt of squre mtr Let deote the mtr The j tr
lso where det det the determt of j j j j cofctor of j the determt of the mtr th th fter deletg row d j col. j
Some roertes. I, tr I. B B, tr B tr B 3. 4. f 0 or 0
Secl Tes of Mtrces. Orthogol mtrces mtr s orthogol f PˊP = PPˊ = I I ths cses P - =Pˊ. lso the rows (colums) of P hve legth d re orthogol to ech other
Secl Tes of Mtrces (cotued). Postve defte mtrces smmetrc mtr,, s clled ostve defte f: for ll 0 smmetrc mtr,, s clled ostve sem defte f: 0 for ll 0 0
Theorem The mtr s ostve defte f 0, 0, 0,, 0 3 33 3 3 3 3 3 d,,, where
Secl Tes of Mtrces (cotued) 3. Idemotet mtrces smmetrc mtr, E, s clled demotet f: E E Idemotet mtrces roject vectors oto ler subsce E E E E E
Egevectors, Egevlues of mtr
Defto Let be mtr Let d be such tht wth 0 the s clled egevlue of d d s clled egevector of d
Note: I 0 If I 0 the I 0 0 thus I 0 s the codto for egevlue.
det = 0 I = oloml of degree. Hece there re ossble egevlues,,
Thereom If the mtr s smmetrc wth dstct egevlues,,,, wth corresodg egevectors,, ssume the 0,, 0 PDP
The Geerlzed Iverse of mtr
Defto B (deoted b - ) s clled the geerlzed verse (Moore Perose verse) of f. B =. BB = B 3. (B)' = B 4. (B)' = B Note: - s uque
Hece B = B B = B B B = B (B ) ' (B ) ' = B B ' ' B ' ' = B B ' ' = B B = B B B = (B )(B )B = (B ) ' (B ) ' B = ' B ' ' B ' B = ' B ' B = (B ) ' B = B B = B The geerl soluto of sstem of Equtos The geerl soluto b I where z b b I z s rbtrr
Let C be q mtr of rk k < m(,q), the C = B where s k mtr of rk k d B s k q mtr of rk k the C B BB
The Geerl Ler Model,, Let X β o dstrbut, hs where I 0 ε ε Xβ N
Geometrcl terretto of the Geerl Ler Model X β,, Let o dstrbut, hs where I 0 ε ε Xβ N X Xβ μ les the ler sce sed b the colums of E
Estmto The Geerl Ler Model
the Norml Equtos
X X Xβ The Norml Equtos, where X
Soluto to the orml equtos X ˆ Xβ X ˆ β XX X If the mtr X s of full rk.
Estmte of ˆ ˆ Xβ ˆ Xβ I X XX ˆ Xβ X
Proertes of The Mmum Lkelhood Estmtes Ubsedess, Mmum Vrce
Ubsedess ˆ β XX X E E XX XE c β c β ˆ ˆ E E cβ E XX XXβ β ˆ s ˆ Xβ ˆ Xβ ˆ s s ubsed estmtor of.
Dstrbutol Proertes Lest squre Estmtes (Mmum Lkeldood estmtes)
The Geerl Ler Model d X X X X I s. X X X X X X β where ˆ. B X X X X I Now s U X X X X I B where I Xβ, ~ N The Estmtes
Theorem. 0 wth, ~. s U, ~ ˆ. XX β β N re deedet d ˆ 3. s β
The Geerl Ler Model wth tercet
0,, Let X β o dstrbut, hs.e. I Xβ N o dstrbut, hs where I 0 ε ε Xβ N The mtr formulto (tercet cluded) The the model becomes Thus to clude tercet dd etr colum of s the desg mtr X d clude the tercet the rmeter vector
The Guss-Mrkov Theorem mortt result the theor of Ler models Proves otmlt of Lest squres estmtes more geerl settg
The Guss-Mrkov Theorem ssume E Xβ d vr I Cosder the lest squres estmte of d ˆ cβ β cx X ˆ XX X X c, ubsed ler estmtor of β Let b b b b β deote other ubsed ler estmtor of the vr ˆ b vrc β c β
Hothess testg for the GLM The Geerl Ler Hothess
Testg the Geerl Ler Hotheses The Geerl Ler Hothess H 0 : h + h + h 3 3 +... + h = h h + h + h 3 3 +... + h = h... h q + h q + h q3 3 +... + h q = h q where h h, h 3,..., h q d h h, h 3,..., h q re kow coeffcets. I mtr otto H q β h q
Testg h Hβ : H 0 Xβ Xβ h Hβ H H X X h Hβ ˆ ˆ ˆ ˆ test sttstc q F q F F H, f Reject 0
ltertve form of the F sttstc F RSS H 0 RSS RSS q RSS H Resdul Sum of Squres ssumg H 0 0 RSS Resdul Sum of Squres ot ssumg H 0
Cofdece tervls, Predcto tervls, Cofdece Regos Geerl Ler Model
c X X c c β ˆ s t Oe t tme ( )00 % cofdece tervl for β c ( )00 % cofdece tervl for d. / / to s s / / to s s
Multle Cofdece Itervls ssocted wth the test H 0 : Hβ h Theorem: Let H be q mtr of rk q. Cosder chβ for ll c the chβ c q, s ch XX ˆ qf H c for ll form set of ( )00 % smulteous cofdece tervl for chβ for ll c c