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Unformatted text preview: Stat 7249: HW 1 solutions 1. a) The likelihood function of exponential family is f ( y, , ) = exp y b ( ) a ( ) + c ( y, ) Then the generating function can be derived as M y ( t ) = E ( e ty ) = Z e ty exp y b ( ) a ( ) + c ( y, ) dy = Z exp y ( + a ( ) t ) b ( + a ( ) t ) a ( ) + c ( y, ) exp b ( + a ( ) t ) b ( ) a ( ) dy = exp b ( + a ( ) t ) b ( ) a ( ) We have K y ( t ) = log M y ( t ) = b ( + a ( ) t ) b ( ) a ( ) b) K y ( t ) = [ b [ + a ( ) t ] a ( )] /a ( ) K 00 y ( t ) = b 00 [ + a ( ) t ] a ( ) K 000 y ( t ) = b 000 [ + a ( ) t ] a 2 ( ) Then K 1 = K y (0) = b (0) , , K 2 = K 00 y (0) = a ( ) b 00 ( ) , , K 3 = K 000 y (0) = a 2 ( ) b 000 ( ) Taking derivatives of the moment generating function with respect to t , we have M y = e K y K y M 00 y = e K y K 00 y + ( K y ) 2 e K y M 000 y = e K y ( t ) K 000 y + K 00 y ( K y ) e K y + 2 e K y K y K 00 y...
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 Spring '08
 Daniels

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