TakeTakeTakeTake----homehomehomehome test test test test Mathematical Statistics IIMathematical Statistics IIMathematical Statistics IIMathematical Statistics II Due: Monday May 4th, 2015 @ 5:00pm Sharp

. You are supposed to work on these problems with NO help from any other sources except your notes and the assigned texts for this course

. In case, your instructor finds some similarity between your work and the other classmates, you may get No credit on this portion of the test.

. I strongly suggest you to write your answers in your own words.

. To get the full credit, show all your work with clear details

. When you staple your work, make sure all pages are in order

. Print this take-home test and use it as the cover sheet of your work

. No electronic submission is accepted

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Problem 1:

Let !" ,!% be a random sample from Normal(&, &'),where & > + is the unknown parameter. 1. Find a sufficient statistic for & 2. Calculate the Fisher information for the sample 3. Derive the Cramer-Rao lower bound for the variance of any unbiased estimators of & 4. Is estimator !, of & efficient (Why?) 5. Is !, asymptotically efficient (Why?) 6. Find estimator of & by the method of moment

Problem 2:

Let !- = /!" ,!%) and 0- = /0" , 0%) be two independent random samples from Exponential(&) and Exponential ("&)respectively, where & > + is the unknown parameter to be estimated from sample (X, Y).

1. Find the minimal sufficient statistic 2. Is this minimal sufficient complete? 3. Find MLE of the &

Problem 3:

Let !" ,!% be a random sample from Gamma /1, 2) whre & = /1, 2) is the unknown parameter. Find UMVUE of 12

Problem 4:

Let !", !',!3 denote three independent observations from a distribution density: 4/5; &) = /" + &)5&+ 5 " 1. Want to test 9+:& = ";