HW2solutions - STA 131B HW2 7.5 8 The likelihood function...

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STA 131B HW2 7.5 8. The likelihood function is f n ( x | θ ) = ( exp( - n i =1 x i ) for min( x 1 , . . . , x n ) > θ 0 otherwise. (a) For each value of x , f n ( x | θ ) will be a maximum when θ is made as large as possible subject to the strict inequality θ < min( x 1 , . . . , x n ). Therefore, the value θ = min( x 1 , . . . , x n ) cannot be used and there is no MLE. (b) Change the pdf of X i to the equivalent f ( x | θ ) ( e θ - x for x θ 0 for x < θ (only the inequality sign is changed), then the likelihood function is f n ( x | θ ) = ( exp( - n i =1 x i ) for min( x 1 , . . . , x n ) θ 0 otherwise. The likelihood function f n ( x | θ ) will be nonzero for θ min( x 1 , . . . , x n ) and the MLE will be ˆ θ = min( x 1 , . . . , x n ). 10. The likelihood function is f n ( x | θ ) = 1 2 n exp ( - n X i =1 | x i - θ | ) . Therefore, the MLE of θ will be the value that minimizes n i =1 | x i - θ | . ˆ θ = the sample median of x i is one of the solutions ( non-unique ) to this minimization problem. To see this: | x i - θ | ∂θ = - 1 , if θ < x i 1 , if θ > x i does not exist , if θ = x i ∂l ( θ ) ∂θ = n X i =1 | x i - θ | ∂θ = X x i 1 + X x i - 1 = - # { x i : x i < θ } + # { x i : x i > θ } , if θ 6 = x i for all i, 1
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where # { x i : x i < θ } means the number of x i s that are smaller than θ . ∂l ( θ ) ∂θ > 0 ( l ( θ ) increasing) when there are more x i lying left to θ , and is positive ( l ( θ ) decreasing) when more x i are lying right to θ , so the maximum of l ( θ ) is taken when there are equal number of
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