Exercises January 21
STA 4508S (Spring, 2014)
Suppose Yi = (Yi1 , . . . , Yid ) ∼ N (0, R), with Rij = 1, if i = j; ρ, if i 6= j. 1. Show that the pairwise composite log-likelihood function takes the form c`(ρ; y1 , . . . , yn ) = −
nm(m − 1) m−1+ρ log(1 − ρ2 ) − SSw 4 2(1 − ρ2 ) (m − 1)(1 − ρ) SSb − 2(1 − ρ2 ) m
and the full log-likelihood function takes the form `(ρ; y1 , . . . , yn ) = −
where SSw =
n(m − 1) n log(1 − ρ) − log{1 + (m − 1)ρ} 2 2 1 1 SSb − SSw − 2(1 − ρ) 2{1 + (m − 1)ρ} m
n X m X
2
(yis − y¯i. ) ,
i=1 s=1
SSb =
n X
yi.2 .
i=1
2. Find an expression for the Godambe information function G(ρ).
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