Dr. Hackney STA Solutions pg 151

Dr. Hackney STA Solutions pg 151 - Second Edition 9-9 an IG...

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Second Edition 9-9 an IG ( n + a, ( y + 1 b ) - 1 ) . The Bayes HPD region is of the form { λ : π ( λ | y ) k } , which is an interval since π ( λ | y ) is unimodal. It thus has the form { λ : a 1 ( y ) λ a 2 ( y ) } , where a 1 and a 2 satisfy 1 a 1 n + a +1 e - 1 a 1 ( y + 1 b ) = 1 a 2 n + a +1 e - 1 a 2 ( y + 1 b ) . b. The posterior distribution is IG((( n - 1) / 2)+ a, ((( n - 1) s 2 / 2)+1 /b ) - 1 ). So the Bayes HPD region is as in part a) with these parameters replacing n + a and y + 1 /b . c. As a 0 and b → ∞ , the condition on a 1 and a 2 becomes 1 a 1 (( n - 1) / 2)+1 e - 1 a 1 ( n - 1) s 2 2 = 1 a 2 (( n - 1) / 2)+1 e - 1 a 2 ( n - 1) s 2 2 . 9.29 a. We know from Example 7.2.14 that if π ( p ) beta( a,b ), the posterior is π ( p | y ) beta( y + a,n - y + b ) for y = x i . So a 1 - α credible set for p is: { p : β y + a,n - y + b, 1 - α/ 2 p β y + a,n - y + b,α/ 2 } . b. Converting to an F distribution, β c,d = ( c/d ) F 2 c, 2 d 1+( c/d ) F 2 c, 2 d , the interval is y + a n - y + b F 2( y + a ) , 2( n - y + b ) , 1 - α/ 2 1 + y + a n - y + b F 2( y + a ) , 2( n - y + b ) , 1 - α/ 2 p y + a n - y + b F 2( y + a ) , 2( n - y + b ) ,α/ 2 1 + y + a n - y + b F 2( y + a ) , 2( n - y + b ) ,α/ 2 or, using the fact that F m,n = F - 1 n,m , 1 1 + n - y + b y + a F 2( n - y + b ) , 2( y + a ) ,α/ 2 p y + a n - y + b F 2( y + a ) , 2( n + b ) ,α/ 2 1 + y + a n - y + b F 2( y + a ) , 2( n - y + b ) ,α/ 2 . For this to match the interval of Exercise 9.21, we need x = y and Lower limit: n - y + b = n - x
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