FRE6091_HW1a_solutions

FRE6091_HW1a_solutions - Solutions Question 1 Yes Any...

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Solutions Question 1 Yes. Any function of data is a statistic. Question 2 Given a random sample u1D44B 1 , ..., u1D44B u1D45B from a population distribution with mean u1D707 and variance u1D70E 2 : First note that a random sample means the components are all independent . Define the statistic u1D447 = ( u1D44B 1 + u1D44B 2 ) / 2 Then (i) u1D438 [ u1D447 ] = ( u1D707 + u1D707 ) / 2 = u1D707 so u1D447 is an unbiased estimator of u1D707 . (ii) u1D449 ( u1D447 ) = (1 / 2) 2 ( u1D449 ( u1D44B 1 ) + u1D449 ( u1D44B 2 )) = u1D70E 2 / 2. There is no covariance because of independence. Since the variance of u1D447 does not decrease to zero as the sample size increases, u1D447 is not a consistent estimator of u1D707 . Question 3 u1D438 ( u1D447 ) = u1D703 = u1D438 ( u1D44Eu1D447 + u1D44F ) = u1D44Eu1D703 + u1D44F since u1D44E and u1D44F are constants. Now u1D438 ( u1D447 2 ) = u1D449 ( u1D447 ) + u1D438 ( u1D447 ) 2 = u1D449 ( u1D447 ) + u1D703 2 . For an unbiased estimator of u1D703 2 we require u1D438 ( u1D447 2 ) = u1D703 2 which is true only in the degenerate case when u1D449 ( u1D447 ) = 0. Question 4 The joint pdf of a random sample u1D44B 1 , ..., u1D44B u1D45B from the uniform distribution on ( u1D703 1 , u1D703 2 ) is
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