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P Set 7 - Problem Set 7 MGTECON 603 2009 MGTECON 603...

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Problem Set 7, MGTECON 603, 2009 1 MGTECON 603, Econometrics I Jules H. van Binsbergen Fall 2009 Stanford GSB PROBLEM SET 7 Due: Monday November 30 th 1. Let the random variable X have probability density function f X ( x ; θ ) = (1 ) exp( x/θ ) for x > 0. Consider the simple hypothesis H 0 : θ = 2 and the alternative hypothesis H 1 : θ = 4. Let X 1 , X 2 denote a random sample of size two from this distribution. Show that the best test of H 0 against H 1 may be carried out by use of the statistic X 1 + X 2 . Find the optimal critical region for α = 0 . 1. 2. Let X 1 , . . . , X 10 be a random sample of size 10 from a normal distribution with mean zero and variance σ 2 . Find a best critical region for testing H 0 : σ 2 = 1 against H 1 : σ 2 = 2 of size α = 0 . 05. Is this also a best test of the same null hypothesis against the alternative hypothesis H 1 : σ 2 = 3? And against the alternative hypothesis H 1 : σ 2 > 1? 3. Let X 1 , X 2 , . . . , X N be a random sample from a distribution with pdf f X ( x ; θ ) = θx θ - 1 for 0 < x < 1. Find the form of the best critical region for testing
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