# hw1.sol - Stat 7249 HW 1 solutions 1 a The likelihood...

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Unformatted text preview: Stat 7249: HW 1 solutions 1. a) The likelihood function of exponential family is f ( y, θ, φ ) = exp yθ- b ( θ ) a ( φ ) + c ( y, θ ) Then the generating function can be derived as M y ( t ) = E ( e ty ) = Z e ty exp yθ- b ( θ ) a ( φ ) + c ( y, φ ) dy = Z exp y ( θ + a ( φ ) t )- b ( θ + a ( φ ) t ) a ( φ ) + c ( y, φ ) exp b ( θ + a ( φ ) t )- b ( θ ) a ( φ ) dy = exp b ( θ + a ( φ ) t )- b ( θ ) a ( φ ) We have K y ( t ) = log M y ( t ) = b ( θ + a ( φ ) t )- b ( θ ) a ( φ ) b) K y ( t ) = [ b [ θ + a ( φ ) t ] a ( φ )] /a ( φ ) K 00 y ( t ) = b 00 [ θ + a ( φ ) t ] a ( φ ) K 000 y ( t ) = b 000 [ θ + a ( φ ) t ] a 2 ( φ ) Then K 1 = K y (0) = b (0) , , K 2 = K 00 y (0) = a ( φ ) b 00 ( θ ) , , K 3 = K 000 y (0) = a 2 ( φ ) b 000 ( θ ) 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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hw1.sol - Stat 7249 HW 1 solutions 1 a The likelihood...

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