Scan_Ch0 - %NUS\li’fl)’:1‘1'3'213LZ” V‘"...

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Unformatted text preview: %NUS \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) / - Chi-square 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 - F-Distribution, 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 = p-dimensional 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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