Test_2_2009 - and S 2 . Note that and cS are unbiased...

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Department of Economics Professor Dale J. Poirier University of California, Irvine December 8, 2009 FINAL EXAM ECON 220A Statistics and Econometrics I (open book) Directions : You must answer each of the following questions. Points (out of 100) are allocated as noted to the left of each question. Allocate your time according to these points. To receive any partial credit, you must show your work. 1. Suppose that the random variables Y 1 and Y 2 have the joint density function f(y 1 , y 2 ) = 3( )/2, for 0 # y 1 # 1 and 0 # y 2 # 1. (5) (a) Are Y 1 and Y 2 exchangeable? (5) (a) Are Y 1 and Y 2 independent? (20) 2. Suppose X 1 and X 2 are i.i.d. with p.d.f. Is W = R n(X 1 )/ R n(X 2 ) an ancillary statistic? 3. Let Y t (t =1, 2, . .., T) be i.i.d. N( 2 , 2 2 ), 2 > 0. Denote the usual sample mean and variance estimators by
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Unformatted text preview: and S 2 . Note that and cS are unbiased estimators of 2 , where (5) (a) Consider the estimator where d is a constant. Is an unbiased estimator of 2 ? (10) (b) Find the value of d for which the MSE is minimized. (5) (c) Is [ , S 2 ] N a sufficient statistic for 2 ? (10) (d) Is [ , S 2 ] N a complete statistic? 2 4. Suppose Z = [Z 1 , Z 2 , Z 3 ] N has the trinomial distribution M 3 (T, 2 ) with p.m.f. where 2 = [(1 - " ) 2 , 2 " (1 - " ), " 2 ] N and " > 0. (5) (a) What is the log-likelihood function for " ? (10) (b) Find the MLE of " . (5) (c) Find the Hessian for " . (10) (d) Find the information for " . (10) 5. Attach your answer to Question 2 of your computer homework assignment....
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This note was uploaded on 12/12/2010 for the course ECON ECON 220A taught by Professor Dalej.poirier during the Fall '10 term at UC Irvine.

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Test_2_2009 - and S 2 . Note that and cS are unbiased...

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