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Dr. Hackney STA Solutions pg 90

Dr. Hackney STA Solutions pg 90 - 6-4Solutions Manual for...

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Unformatted text preview: 6-4Solutions Manual for Statistical InferenceThe last ratio does not depend onθ. The other terms are constant as a function ofθif andonly ifn=nandx=y. So (X,N) is minimal sufficient forθ. BecauseP(N=n) =pndoes not depend onθ,Nis ancillary forθ. The point is that althoughNis independent ofθ, the minimal sufficient statistic containsNin this case. A minimal sufficient statistic maycontain an ancillary statistic.b.EXN=EEXNN= E1NE (X|N)= E1NNθ= E(θ) =θ.VarXN=VarEXNN+ EVarXNN= Var(θ) + E1N2Var (X|N)=0 + ENθ(1-θ)N2=θ(1-θ)E1N.We used the fact thatX|N∼binomial(N,θ).6.13 LetY1= logX1andY2= logX2. ThenY1andY2are iid and, by Theorem 2.1.5, the pdf ofeach isf(y|α) =αexp{αy-eαy}=11/αexpy1/α-ey/(1/α),-∞< y <∞.We see that the family of distributions ofYiis a scale family with scale parameter 1/α. Thus,by Theorem 3.5.6, we can writeYi=1αZi, whereZ1andZ2are a random sample fromf(z|1)....
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