P Set 6 - 1 MGTECON 603 Econometrics I Fall 2009 PROBLEM...

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1 MGTECON 603, Econometrics I Jules H. van Binsbergen Fall 2009 Stanford GSB PROBLEM SET 6 Due: Monday November 16 th 1. Let X 1 , X 2 , . . . , X N represent a random sample from a Poission distribution with arrival rate λ . (a) Find the maximum likelihood estimator for λ and its asymptotic distribution. (b) Suppose we are interested in the probability of a count of zero, θ = Pr ( X = 0) = exp( - λ ). Find the maximum likelihood estimator for θ and its asymptotic distribution. (c) Show that X i is a sufficient statistic. (d) Find an unbiased estimator for θ . (Hint: define Y as the indicator for the event X = 0. Then E [ Y ] = Pr ( X = 0).) Is this estimator a function of the sufficient statistic? Modify the estimator to make sure it is a function of the sufficient statistic. (e) How does the variance of the unbiased estimator compare to the asymptotic vari- ance of the maximum likelihood estimator? 2. 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 asymptotic variance
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