Hw11Sol - SIEO 3600 (IEOR Majors) Solution of Assignment...

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Unformatted text preview: SIEO 3600 (IEOR Majors) Solution of Assignment #11 Introduction to Probability and Statistics April 20, 2010 Solution of Assignment #11 ( Maximum likelihood estimator and confidence interval ) 1. Let X 1 ,...,X n be a sample from the distribution whose density function is f X ( x ) = e- ( x- ) , x , , otherwise . Determine the maximum likelihood estimator of . Hint: You cannot match the derivative of log L ( ) to 0 in this case. Solution: Step 1: We write out the likelihood function L ( ) = f ( x 1 ,...,x n ) = f ( x 1 ) f ( x n ) = e- ( x 1- ) e- ( x n- ) = e- P n i =1 x i + n . Step 2: We take the logrithm of L ( ) log L ( ) =- n X i =1 x i + n. Step 3: We want to solve the following maximization problem: max log L ( ) =- n X i =1 x i + n such that min( x 1 ,...,x n ) . (1) Note that constraint (1) make sense because the PDF is 0 for all x < , therefore observations x 1 ,...,x n all have to be greater than . To solve this optimiztion problem in Step 3, it is easy to see that we cannot match the derivative of log L ( ) with respect to (that is n ) and match it to 0. Indeed, we see that the objective function is just a linear function of , therefore the bigger is, the bigger the objective value is. Because of constraint (1), the biggest would be the minimun of x i . Hence, the MLE of is = min( X 1 ,...,X n )....
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This note was uploaded on 10/03/2011 for the course SIEO W3600 taught by Professor Yunanliu during the Spring '10 term at Columbia.

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Hw11Sol - SIEO 3600 (IEOR Majors) Solution of Assignment...

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