fexamAS-f09 - Economics 506 FINAL EXAM Total Points...

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Economics 506 FALL 2009 FINAL EXAM Total Points: 200 (SOLUTION) Name: __Ali Toossi_________________________ e- mail:_______________________ 1) [Total points: 80] Consider the following density function = - - elsewhere 0 0 , 0 1 ) ( 1 y e ry y f r y r θ where r is a known positive constant. a) [10 points] use the method of transformation to find the density function of U = Y r u r u r r r r Y U r r r e u r e u r u r u f u f u r dr dY u Y Y U r r - - - - - = = = = = = 1 1 ) ( 1 1 ) ( ) ( 1 1 1 ) ( 1 1 1 1 1 1 1 1 1 Thus distribution of U is exponential for U >0 b)[10 points] Let Y 1 , Y 2 , …, Y n denote a random sample from the density function given above. Find a sufficient statistics for θ. The likelihood function is ( 29 ( 29 = - = - θ - θ = θ n i r i r n i i n n y y r L 1 1 1 / exp ) ( . a sufficient statistic for θ is = n i r i Y 1 . c) [10 points] Use method of moment generating function to find the distribution of the sufficient statistics you found in part (b). We know from part (a) that r Y has exponential, therefore each r i Y has exponential distribution. Now: n Y Y Y Y t t t t t M t M t M t M r n r r r i - - - - - = - - - = = ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) ( 1 1 1 2 1 . Therefore the distribution of = n i r i Y 1 is Gamma with parameters n & θ. (Use the back side if you need to) 1
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Economics 506 FALL 2009 FINAL EXAM d) [10 points] Let Y 1 , Y 2 , …, Y n denote a random sample from the density function given above. Find the maximum likelihood estimator (MLE) of θ The log–likelihood is ( 29 = = θ - - + + θ - = θ n i r i n i i y y r r n n L 1 1 / ln ) 1 ( ln ln ) ( ln . By taking a derivative w.r.t. θ and equating to 0, we find = = θ n i r i n Y 1 1 ˆ . e) [5 points] Is the MLE you found in part (d) a MVUE (minimum variance unbiased estimator) for θ? Note that θ ˆ is a function of the sufficient statistic. We showed that Y r has exponential distribution thus θ = ) ( r Y E , therefore θ = = = = = = = n i n i r i n i r i n n Y E n Y E E 1 1 1 1 1 ) ( 1 ) ( ) ˆ ( This means that θ ˆ is also unbiased. Hence it is MVUE for θ. f) [10 points] Find the minimum variance of an unbiased estimator of , using Rao- Cramer inequality? Confirm that it is the same as the variance of the MLE you found in part (d). n
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fexamAS-f09 - Economics 506 FINAL EXAM Total Points...

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