P Set 4 - Problem Set 4 MGTECON 603 2009 MGTECON 603...

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Problem Set 4, MGTECON 603, 2009 1 MGTECON 603, Econometrics I Jules H. van Binsbergen & Xue Ying Fall 2009 Stanford GSB PROBLEM SET 4 Due: Monday October 26 rd in the CA session 1. Let X 1 , X 2 , . . . , X N be a set of N < independent random variables with Gumbel distributions. The cdf of X i is given by F ( x ) = exp ( - exp ( - x - γ + μ i )) where γ = 0 . 5775 is known as Euler’s constant and μ i is a parameter that is different for each i . The mean of this distribution is given by μ i . (a) What is the distribution of Y 1 = max { X 1 , ..., X N } ? (b) What is Pr ( X 1 max { X 2 , ..., X N } )? 2. Let X 1 , X 2 , . . . be a sequence of independent random variables with a unit exponential distribution. Use the delta method to find an approximate normal distribution for 1 / X n . 3. Find an example where the sequence X 1 , X 2 , . . . converges almost surely but not in quadratic mean. (You are allowed to use any book you like, but explain it well!!!) 4. Show that if the sequence X 1 , X 2 , . . . converges to μ in probability, then it converges to μ in distribution. (You are allowed to use any book you like, but explain it well!!!)
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