Tutprac6 - 620-202 Tutorial/Computing Labo-ratory 6.1(6.14-1 LetX1,Xnbe a random sample fromN(θ,σ2 whereσ2is known(a Show thatY=(X1 X2/2 is an

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Unformatted text preview: 620-202 Tutorial/Computing Labo-ratory 6.1. (6.14-1) LetX1,...,Xnbe a random sample fromN(θ,σ2), whereσ2is known.(a) Show thatY= (X1+X2)/2 is an unbiased estimator ofθ.(b) Find the Rao-Cr´amer lower bound for the variance of an unbiased estimatorofθ.(c) The efficiency of an estimator is the ratio of the Rao-Cr´amer lower boundto the variance of the estimator. What is the efficiency of the estimator in(a)?2. (6.14-3) LetX1,...,Xnbe a random sample fromN(μ,θ), whereμis known.(a) Show the maximum likelihood estimator ofθisˆθ=n-1∑ni=1(Xi-μ)2.(b) Determine the Rao-Cr´amer lower bound.(c) What is the approximate distribution ofˆθ?(d) What is the exact distribution ofnˆθ/θ? Can you use your knowledge of thisdistribution to determine ifˆθattains the Rao-Cr´amer lower bound?3. SupposeX∼U(0,θ). We take one observation onX.(a) Find the method of moments estimator ofθ.(b) Find the maximum likelihood estimator ofθ....
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This note was uploaded on 10/05/2009 for the course STATS 620-202 taught by Professor R during the One '09 term at University of Melbourne.

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Tutprac6 - 620-202 Tutorial/Computing Labo-ratory 6.1(6.14-1 LetX1,Xnbe a random sample fromN(θ,σ2 whereσ2is known(a Show thatY=(X1 X2/2 is an

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