411_hw05_soln

411_hw05_soln - Stat 411 Homework 05 Solutions 1. (a) The...

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Unformatted text preview: Stat 411 Homework 05 Solutions 1. (a) The likelihood function looks like L ( 1 , 2 ) = n Y i =1 f ( X i ) - n/ 2 2 e- (1 / 2 2 ) n i =1 (log X i- 1 ) 2 . Taking a natural logarithm gives ( 1 , 2 ) = const- n log 2 2- 1 2 2 n X i =1 (log X i- 1 ) 2 . Differentiating with respect to 1 and 2 separately and setting equal to zero gives the likelihood equations: 0 = 1 ( 1 , 2 ) = 1 2 n X i =1 (log X i- 1 ) , 0 = 2 ( 1 , 2 ) =- n 2 2 + 1 2 2 2 n X i =1 (log X i- 1 ) 2 . The solution to the first equation is 1 = 1 n n i =1 log X i , irrespective of the value of 2 . With the solution 1 plugged in to the second equation, the solution for 2 is clearly 2 = 1 n n i =1 (log X i- 1 ) 2 . This is just like if we had originally made a transformation Y = log X and did the calculations with Y 1 ,...,Y n iid N ( 1 , 2 ); that is, the MLEs would be 1 = Y and 2 = 1 n n X i =1 ( Y i- Y ) 2 , Y i = log X i , i = 1 ,...,n. (b) Towards the Fisher information matrix, we need various derivatives of log f ( x ): 1 log f ( x ) = log x- 1 2 2 log f ( x ) =- 1 2 2 + (log x- 1 ) 2 2 2 2 2 2 1 log f ( x ) =- 1 2 2 2 2 log f ( x ) = 1 2 2 2- (log x- 1 ) 2 3 2 2 1 2 log f ( x ) =- log x- 1 2 2 ....
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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411_hw05_soln - Stat 411 Homework 05 Solutions 1. (a) The...

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