Written Problems l. Let Yı, Ya, Ya be a random sample from an Exponential distribution with...

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Written Problems l. Let Yı, Ya, Ya be a random sample from an Exponential distribution with density function f(y)-Te-3, y > 0. Find the MSE of each of the following estimators of θ: (a)-互华 (c) θ=F 2

math Statistics-And-Probability

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Answer 1

Answer #1

Since $Y_{1},Y_{2},Y_{3}$ are exponential distribution so we have

$E(Y_{1})= heta$ , $E(Y_{2})= heta$ and $E(Y_{3})= heta$

$V(Y_{1})= heta^{2}$ , $V(Y_{2})= heta^{2}$ and $V(Y_{3})= heta^{2}$

(a)

$E(hat{ heta})=Eleft ( rac{Y_{1}+Y_{2}}{2} ight )=rac{1}{2}left [Eleft ( Y_{1} ight )+Eleft ( Y_{2} ight ) ight ]=rac{1}{2}cdot 2 heta= heta$

So it unbiased.

$V(hat{ heta})=Vleft ( rac{Y_{1}+Y_{2}}{2} ight )=rac{1}{4}left [Vleft ( Y_{1} ight )+Vleft ( Y_{2} ight ) ight ]=rac{1}{2}cdot 2 heta^{2}=rac{1}{2} heta^{2}$

So MSE will be

e2 e2 ĮE(9)-ตุใーー旧ー012 = MSE0 = Var@)

(b)

E(θ) = E

So it unbiased.

$V(hat{ heta})=Vleft ( rac{Y_{1}+2Y_{2}}{3} ight )=rac{1}{9}left [Vleft ( Y_{1} ight )+4Vleft ( Y_{2} ight ) ight ]=rac{5}{9} heta^{2}$

So MSE will be

MSE0 = Var@) |E(θ) _ θ]2ーー+旧ー012ーー 5g2 5θ2

(c)

$E(hat{ heta})=E(ar{Y})=Eleft ( rac{Y_{1}+Y_{2}+Y_{3}}{3} ight )=rac{1}{3}left [E left ( Y_{1} ight )+E( Y_{2})+E( Y_{3}) ight ]=rac{1}{3}cdot 3 heta= heta$

So it unbiased.

392ー -θ2

So MSE will be

e2 e2 MSE; = Var@) + |E( )-ตุใ =ー+ [θ _ θ]2 . 3

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Answer 2

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