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 on. The other terms are constant as a function ofif andonly ifn=nandx=y. So (X,N) is minimal sufficient for. BecauseP(N=n) =pndoes not depend on,Nis ancillary for. The point is that althoughNis independent of, the minimal sufficient statistic containsNin this case. A minimal sufficient statistic maycontain an ancillary statistic.b.EXN=EEXNN= E1NE (X|N)= E1NN= E() =.VarXN=VarEXNN+ EVarXNN= Var() + E1N2Var (X|N)=0 + EN(1-)N2=(1-)E1N.We used the fact thatX|Nbinomial(N,).6.13 LetY1= logX1andY2= logX2. ThenY1andY2are iid and, by Theorem 2.1.5, the pdf ofeach isf(y|) =exp{y-ey}=11/expy1/-ey/(1/),-< y <.We see that the family of distributions ofYiis a scale family with scale parameter 1/. Thus,by Theorem 3.5.6, we can writeYi=1Zi, whereZ1andZ2are a random sample fromf(z|1)....
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