Dr. Hackney STA Solutions pg 92

Dr. Hackney STA Solutions pg 92 - 6-6Solutions Manual for...

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Unformatted text preview: 6-6Solutions Manual for Statistical Inferencefrom a uniform(1,2) population, thenX1=θZ1,...,Xn=θZnis a random sample from auniform(θ,2θ) population, andX(1)=θZ(1)andX(n)=θZ(n). SoX(1)/X(n)=Z(1)/Z(n), astatistic whose distribution does not depend onθ. Thus, as in Exercise 6.10, (X(1),X(n)) is notcomplete.6.24 Ifλ= 0, Eh(X) =h(0). Ifλ= 1,Eh(X) =e-1h(0) +e-1∞Xx=1h(x)x!.Leth(0) = 0 and∑∞x=1h(x)x!= 0, so Eh(X) = 0 buth(x)6≡0. (For example, takeh(0) = 0,h(1) = 1,h(2) =-2,h(x) = 0 forx≥3 .)6.25 Using the fact that (n-1)s2x=∑ix2i-n¯x2, for any (μ,σ2) the ratio in Example 6.2.14 canbe written asf(x|μ,σ2)f(y|μ,σ2)= exp"μσ2Xixi-Xiyi!-12σ2Xix2i-Xiy2i!#.a. Do part b) first showing that∑iX2iis a minimal sufficient statistic. Because(∑iXi,∑iX2i)is not a function of∑iX2i, by Definition 6.2.11(∑iXi,∑iX2i)is not minimal.b. Substitutingσ2=μin the above expression yieldsf(x|μ,μ)f(y|μ,μ)= exp"Xixi-Xiyi#exp"-12μXix2i-Xiy2i!#....
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