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\li’fl)’ :1‘1'3'213LZ” V‘ " Chagter 0 Basic Prerequisite Knowledge ST3‘I 31 Regression Analysis > 0.1 1. Distributions  Normal Distribution, N01, 02) /  Chisquare Distribution, x2 (n)
— Let U x ZZ. IfZ ~ N(O, 1), then (mummy Cid ~ 1' 1 I U ~ x20)
Definition \
— Let Wm X1 + + X”. IfX, ~ x2(1],i= 1, ..., n,
independently, then
W ~ XZW
Deﬁnition9=
LEV Zi;“'Zh N Nb ‘
ST3131 Re ression Anal sis
g y W: 212+ 221*“ W
' t—Distribution, t(mf day/€94 9% ham“ — IfZ w NU), 1) and V N X2072) independently, then  FDistribution, F (m, n)
— If V ~ szz) and W ~ x201) independently, then F V/m F(
m m w my:
W/n 7‘ I. Wraneécr 9
pmmm' i
ST3131 Regression Analysis \ / 0.3 can not Ohqu 03rde 2. Confidence Interval A
o If § is a point estimate of 9, which follows a
normal. or an approximate normal distribution, :5 6‘9 JV M O] ‘ >
0 then a 100(1 — a)% confidence interval for G is ‘5 . A z
giVen by Wt (V is anknbwn,9§b‘mm JOYEQM‘DD
(9 i t m 5.8.(é) v“  N l 2.
9’1 (Rf/2 (y)
0‘
where s.e.(é) is the standard error (Le. the + rgQié) f and é’Q are mcingMient
estimated standard deviation) of § [0: r V v —~ ‘ x
t) (9 “g m ‘i: i V) p
a 99(5). .—/ — 4‘ M
\ ‘ ‘ kg
(,thfrden Le Miami ( / ST3131 Regression Analysis 04 gamus
\gjwy 3. Review of Matrices 3.1 Notation  A = ((117),: 1, m: 1’ q denotes a p X C] matrlx. €11  _C_l_ = 5 denotes a p—dimensional vector.
at?  A’ = (aﬁ) = transpose ofA.  110 = p X [9 identity matrix.  lp = pdimensional vector of 1’s. ST3131 Regression Analysis 0.5 3. Review of Matrices 3.1 Notation
IfA is a p X p matrix, then
 M] x determinam OfA (or det(A)), 0 A4 : inverse ofA, (i.e.AA‘1 = A‘lA = I .)
PW a WM 2x), mah‘rx ; (0; iiiimitzzﬂi “ii ST3131RegressionAnalysis T ‘ a b .6
dece'rminant <4: ( C d8 NUS
‘ 333;: 3.2 Exgectation X1 Y1
 Let g m 5 and}: m 5 be random vectors
X39 Y}?
5 (X1) 50/1)
 Then 5(5) m 5 and EQ) m 3
503(3)) E(Yp)  HA is a q X p matrix of constants and b is a q X 1
vector of constants, then EMQQ+§FAHXQ+Q ST3131 Regression Analysis 0.7 N us
\Lftg' 3.3 Covariance Matrix  Covariance matrix is defined as 601201,!) —— E [({1— E(£)) (X — E(z))'] .
E (Xi — maim — 1509)) Eiwi—Ecxiiﬁmwmi (5.014% 3/1)). . i=1,...1p;j:1’...’p ST3131 Regression Analysis 0.8 (J4; ‘\ 3.3 Covariance Matrix (Continued)  When X = )_(, Cal/(X, )_Q is called the variance—
Covariance matrix of )_( (or dispersion matrix) and is
denoted by VQQ. W) 3 60mg.) Warm) Cay(X3,X2) maﬁa?)
m Cov(X2,X1) Von/(X2) Cev(X2,Xp)
VartXp) ST3131 Regression Analysis 0.9 3.3 Covariance Matrix (Continued) 0 Note: WA) is asymmetric matrix. (Iv/(>521) I\s 043m o {rmmétrtc Matrix  HA is a p X 19 matrix of constants and b is a p X 1
vector of constants, then Wt gm z_;) :A VQQ A’.
o In particular, WA 2Q 2 A VQQ A’.  Note: Adding a constant vector b to the random
vector AX does not change the variance of A)_(. ST3131 Regression Analysis 010 3.4 Some grogerties of matrices
1. AB i BA Z. {A’}’ m 14% and {A}??? m 32%” 3. AB “1 x 31144 providin A‘1 and B“1 exist
8 ST3131 Regression Analysis 0“ ...
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