hw9 - θ units to the right 5 Let X 1,X 2,X n be a random...

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APPM 4/5520 Problem Set Nine (Due Wednesday, November 30th) 1. Let X 1 , X 2 , . . . , X n be a random sample from the exponential distribution with rate λ . (a) Find the MLE (maximum likelihood estimator) for the “tail probability” P ( X 1 > x ) for some fixed x > 0. (b) Find the asymptotic distribution of your estimator from part (a). 2. Let X 1 , X 2 , . . . , X n be a random sample from the N ( μ, σ 2 ) distribution where μ is known. Find the efficiency of the estimator hatwider σ 2 = S 2 . 3. Let X 1 , X 2 , . . . , X n be a random sample from the Poisson distribution with parameter λ . Verify, from the definition, that S = n i =1 X i is a sufficient statistic for λ . (i.e. Do not use the Factorization Criterion.) 4. Find a sufficient statistic or a set of jointly sufficient statistics for the parameter(s) of (a) the Pareto ( γ ) distribution (b) the N ( μ, σ 2 ) distribution (c) the shifted exponential distribution with rate λ that has been shifted θ units to the right 5. Let X 1 , X 2 , . . . , X
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Unformatted text preview: θ units to the right 5. Let X 1 ,X 2 ,... ,X n be a random sample from a distribution with pdf f ( x ; θ ). Under the “regularity conditions” for the CRLB, (a) Find the asymptotic distribution of n s i =1 ∂ ∂θ ln f ( X i ; θ ) (b) What does 1 n n s i =1 p-∂ 2 ∂θ 2 ln f ( X i ; θ ) P converge in probability to? (Explain.) 6. [Required for 5520 Only] Let X 1 ,X 2 ,... ,X n be a random sample from a distribution with pdf f ( x ; θ ). A statistic M is minimal sufcient for this model if it is su²cient and a function of every other set of su²cient statistics. Suppose that T = t ( v X ) is any statistic. Show that T is minimal su²cient if the following condition holds for any vx and vy . f ( vx ; θ ) f ( vy ; θ ) is independent of θ ⇔ t ( vx ) = t ( vy ) ....
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