Transcription of Math 362, Problem Set 6 - University of Denver
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Math 362, Problem Set 6 - University of Denver
Math 362, Problem Set 6. Due 3/23/11. 1. ( ) Let X1 , X2 , .. , Xn be a random sample from a ( = 3, = ). distribution , 0 < < . Determine the mle of . Answer We have 1 Y P. L( ) = (Xi2 )e Xi / . 3n 2n X X. `( ) = 3n log( ) n log(2) + 2 log(Xi ) Xi / . P. 3n Xi `0 ( ) = + 2.. Solving `0 ( ) = 0, we have 1. = X . 3. 2. ( ) Suppose X1 , .. , Xn are iid with pdf f (x; ) = 2x/ 2 , 0 < x , zero elsewhere. Find (a) The mle for . (b) The constant c so that E[c ] = . (c) The mle for the median of the distribution . Answer: We have that 2n Q.
6. (6.2.7’) Let Xhave a gamma distribution with = 3 and = >0. (a) Find the Fisher information I( ). (b) If X 1;:::;X n is a random sample from this distribution, show that the mle of is an e cient estimator of . (c) What is the asymptotic distribution of p n( ^ )? Note: I changed = 4 in the original problem to = 3 since you
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