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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online