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# notes7s - Supplementary Notes Lemma: Let X = (X1 , ., Xk )...

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Supplementary Notes Lemma : Let X = ( X 1 , ..., X k ) 0 has a density f ( x ). Y = g ( X ) where g is 1-1. Then Y has the pdf f [ g - 1 ( y )] | J | + where | J | = | ∂x ∂y | = ± ± ± ± ± ± ± ± ± ± ∂x 1 ∂y 1 ∂x 1 ∂y 2 . . ∂x 1 ∂y k . . . . . . . . . . ∂x k ∂y 1 ∂x k ∂y k . . ∂x k ∂y k ± ± ± ± ± ± ± ± ± ± . To derive the likelihood function for the AR(1) model given observations z 1 , z 2 , . . . , z n and a t iid N (0 , σ 2 a ) . Recall that the AR(1) model Z t - μ = φ ( Z t - 1 - μ ) + a t , t = 1 , 2 , ..., n The pdf of a t is 1 q 2 πσ 2 a e - ( a 2 t 2 σ 2 a ) , - ∞ < a t < . The joint pdf of a 2 , a 3 , . . . , a n is n Y t =2 1 q 2 πσ 2 a e - ( a 2 t 2 σ 2 a ) = (2 πσ 2 a ) - ( n - 1) 2 e - 1 2 σ 2 a n t =2 a 2 t . Note that Z 2 - μ = φ ( Z 1 - μ ) + a 2 Z 3 - μ = φ ( Z 2 - μ ) + a 3 ..... ..... Z n - μ = φ ( Z n - 1 - μ ) + a n . Condition on Z 1 = z 1 , then the joint pdf of Z 2 , . . . , Z n given Z 1 = z 1 is given by f ( z 2 , . . . , z n | z 1 ) = (2 πσ 2 a ) - ( n - 1)

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## This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.

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notes7s - Supplementary Notes Lemma: Let X = (X1 , ., Xk )...

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