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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 θ ) , x ≥ θ, , otherwise . Determine the maximum likelihood estimator of θ . Hint: You cannot match the derivative of log L ( θ ) to 0 in this case. Solution: Step 1: We write out the likelihood function L ( θ ) = f θ ( x 1 ,...,x n ) = f θ ( x 1 ) ··· f θ ( x n ) = e ( x 1 θ ) ··· e ( x n θ ) = e P n i =1 x i + nθ . Step 2: We take the logrithm of L ( θ ) log L ( θ ) = n X i =1 x i + nθ. Step 3: We want to solve the following maximization problem: max log L ( θ ) = n X i =1 x i + nθ such that θ ≤ min( x 1 ,...,x n ) . (1) Note that constraint (1) make sense because the PDF is 0 for all x < θ , therefore observations x 1 ,...,x n all have to be greater than θ . To solve this optimiztion problem in Step 3, it is easy to see that we cannot match the derivative of log L ( θ ) with respect to θ (that is n ) and match it to 0. Indeed, we see that the objective function is just a linear function of θ , therefore the bigger θ is, the bigger the objective value is. Because of constraint (1), the biggest θ would be the minimun of x i . Hence, the MLE of θ is ˆ θ = min( X 1 ,...,X n )....
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