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STAT3922: Applied Linear Model

创建日期：2024-06-17 16:36:10

所属分类：统计学statistics

标签： STAT3922

Applied Linear Model

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STAT3922: Applied Linear Model (Advanced) Semester 1, 2024

Specifically, consider the linear model y = Xβ+ε, whereX is an n×p full rank design
matrix. Consider the general linear hypothesis test H0 : Lβ = 0 versus H1 : Lβ ̸= 0,
for some q × p matrix L with rank(L) = q < p. Let βˆ = (X⊤X)−1X⊤y denote the least
square estimator under H1 (i.e the unconstrained model). In the Advanced Lecture week
10, we construct
SSH = βˆ⊤L⊤
{
L(X⊤X)−1L⊤
}−1
(Lβˆ)
and build the test based on F =
SSH/q
SSE(H1)/(n− p) , where SSE(H1) = y
⊤(In−H)y is the
SSE for the full model.
Also, we are proving that this test can also be conducted from the full-reduced model
approach. To do it, we fitted the model under H0 using the Lagrange multiplier approach,
showed that the least square estimator for the model under H0 is given by
β˜ = (X⊤X)−1
(
X⊤ −K)y,
where K = L⊤
{
L(X⊤X)−1L⊤
}−1
L(X⊤X)−1X⊤. The residual vector of that model is
e˜ = y − y˜ = (I−P)y, y˜ = Xβ˜ = Py,
where P = X(X⊤X)−1
(
X⊤ −K). Fill in the details for the remaining step:
1. Show that
(i) P is idempotent.
(ii) trace(P) = p− q.
2. Let SSE(H0) = e˜
⊤e˜ be the residuals sum of squares under H0. Show that
SSE(H0)− SSE(H1) = SSH.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 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