# P Set 5 - 1 MGTECON 603 Econometrics I Fall 2009 Jules H...

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1 MGTECON 603, Econometrics I Jules H. van Binsbergen and Ying Xue Fall 2009 Stanford GSB PROBLEM SET 5 Due: Monday November 2 nd 1. Let X 1 , X 2 , . . . , X N represent a random sample from each of the distributions having the following probability density functions. In each case find the maximum likelihood estimator ˆ θ of θ . (a) f ( x ; θ ) = θx θ - 1 , for 0 < x < 1, and zero elswhere, for 0 < θ < . (b) f ( x ; θ ) = (1 / 2) exp( -| x - θ | , for -∞ < x < , and zero elswhere, for -∞ < θ < . 2. Let X 1 , X 2 , . . . , X N be a random sample from the distribution having pdf f X ( x ; θ 1 , θ 2 ) = (1 2 ) exp( - ( x - θ 1 ) 2 ) for θ 1 < x < , and -∞ < θ 1 < , 0 < θ 2 < . Find the maximum likelihood estimators of θ 1 and θ 2 . 3. Let Y 1 , Y 2 be a pair of independent, identically distributed random variables with common density f Y ( y | θ ) = 1 θ exp( - y/θ ) for y > 0 and zero elsewhere, and for some θ > 0. Consider the estimator W λ,δ = λY 1 + δY 2 for fixed values of λ and δ . For which values of λ and δ is W λ,δ unbiased?

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