Let X1 be the sample mean of a random sample of size n from a Norinal(µ, o†) population and X2 be the sample mean of a random sample of the same sizC n from a Normal(jµ, o3) population and the two samples are independent. Note that the two samples have the sane population Imcan i but of #03. Imean Let ji = wX¡ +(1 – w)X2,0

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Let X1 be the sample mcan of a random sample of size n from a Norinal(jµ, o†) population and
X2 be the sample mean of a random sample of the same size n from a Normal(j1, o,) population
and the two samples are independent. Note that the two samples have the same population
Imean ji but o + o3.
Let i = wX1 +(1 – w)X2,0 <w<1 be an cstimator for ji. Is îî unbiased for
(a)
L? Explain.
Find the value of w* in [0,1] so that Var(î) is minimized. Hint: take the
(b)
derivative of Var(ſî) with respect to w and set it equal to 0.
(c)
that all population parameters µ, 07,0% are unknown, does it make sense to use îiª as an
estimator for µ? If ycs, explain why. If no, provide a modification of i that has a similar
structure but makes sense as an estimnator for 1.
Let i := w*X1+(1-w*)X2 where w' is the answer from part b. If we assume
Transcribed Image Text:Let X1 be the sample mcan of a random sample of size n from a Norinal(jµ, o†) population and X2 be the sample mean of a random sample of the same size n from a Normal(j1, o,) population and the two samples are independent. Note that the two samples have the same population Imean ji but o + o3. Let i = wX1 +(1 – w)X2,0 <w<1 be an cstimator for ji. Is îî unbiased for (a) L? Explain. Find the value of w* in [0,1] so that Var(î) is minimized. Hint: take the (b) derivative of Var(ſî) with respect to w and set it equal to 0. (c) that all population parameters µ, 07,0% are unknown, does it make sense to use îiª as an estimator for µ? If ycs, explain why. If no, provide a modification of i that has a similar structure but makes sense as an estimnator for 1. Let i := w*X1+(1-w*)X2 where w' is the answer from part b. If we assume
Expert Solution
Step 1

A statistic t is said to be an unbiased estimate of a parameter θ if Et=θ.

Given

μ^=ω X¯1+1-ωX¯2

Also

EX¯1=EX¯2=μ

Consider,

Eμ^=Eω X¯1+1-ωX¯2=ω*E X¯1+1-ω*EX¯2=ωμ+1-ωμ=ω+1-ωμ=μ

  μ^ is an unbiased estimator of μ.

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