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Savatey [412]
3 years ago
14

HELP PLZZZZZ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Mathematics
1 answer:
Neporo4naja [7]3 years ago
7 0

Answer: I think it’s  A because 8x5=40

Step-by-step explanation:

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Suppose that E(θˆ1) = E(θˆ2) = θ, V(θˆ 1) = σ2 1 , and V(θˆ2) = σ2 2 . Consider the estimator θˆ 3 = aθˆ 1 + (1 − a)θˆ 2. a Show
katen-ka-za [31]

Answer:

Step-by-step explanation:

Given that:

E( \hat \theta _1) = \theta  \ \ \ \ E( \hat \theta _2) = \theta \ \ \ \ V( \hat \theta _1) = \sigma_1^2  \ \ \ \ V(\hat  \theta_2) = \sigma_2^2

If we are to consider the estimator \hat \theta _3 = a \hat  \theta_1 + (1-a)  \hat \theta_2

a. Then, for  \hat \theta_3 to be an unbiased estimator ; Then:

E ( \hat \theta_3) = E ( a \hat \theta_1+ (1-a) \hat \theta_2)

E ( \hat \theta_3) = aE (  \theta_1) + (1-a) E ( \hat \theta_2)

E ( \hat \theta_3) = a   \theta + (1-a)  \theta = \theta

b) If \hat \theta _1 \ \  and  \ \   \hat \theta_2 are independent

V(\hat \theta _3) = V (a \hat \theta_1+ (1-a) \hat \theta_2)

V(\hat \theta _3) = a ^2 V ( \hat \theta_1) + (1-a)^2 V ( \hat \theta_2)

Thus; in order to minimize the variance of \hat \theta_3 ; then constant a can be determined as :

V( \hat \theta_3) = a^2 \sigma_1^2 + (1-a)^2 \sigma^2_2

Using differentiation:

\dfrac{d}{da}(V \ \hat \theta_3) = 0 \implies 2a \ \sigma_1^2 + 2(1-a)(-1) \sigma_2^2 = 0

⇒

a (\sigma_1^2 + \sigma_2^2) = \sigma^2_2

\hat a = \dfrac{\sigma^2_2}{\sigma^2_1+\sigma^2_2}

This implies that

\dfrac{d}{da}(V \ \hat \theta_3)|_{a = \hat a} = 2 \ \sigma_1^2 + 2 \ \sigma_2^2 > 0

So, V( \hat \theta_3) is minimum when \hat a = \dfrac{\sigma_2^2}{\sigma_1^2+\sigma_2^2}

As such; a = \dfrac{1}{2}       if   \sigma_1^2 \ \ =  \ \  \sigma_2^2

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