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h2 - Stat 5102(Geyer Spring 2012 Homework Assignment 2 Due...

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Stat 5102 (Geyer) Spring 2012 Homework Assignment 2 Due Wednesday, February 1, 2012 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 2-1. Suppose X 1 , X 2 , . . . are IID Unif(0 , θ ). As usual X ( n ) denotes the n -th order statistic, which is the maximum of the X i . (a) Show that X ( n ) P -→ θ, as n → ∞ . (b) Show that n ( θ - X ( n ) ) D -→ Exp(1 ) , as n → ∞ . Hints This is a rare problem (the only one of its kind we will meet in this course) when we can’t use the LLN or the CLT to get convergence in probability and convergence in distribution results (obvious because the problem is not about X n and the asymptotic distribution we seek isn’t normal). Thus we need to derive convergence in distribution directly from the characterization as convergence of distribution functions (5101 deck 6, slide 4), that is, X n D -→ X if and only if F n is the DF of X n and F is the DF of X and F n ( x
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