Let X1,X2......X7 denote a random sample from a population having mean μ and variance σ. Consider the following estimators of μ: θ1=X1+X2+......+X7 / 7 θ2= (2X1-X3+X5) / 2 a. Is …

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asked Jun 19, 2021 174k views

1 Answer

Truong Hua · Answer 1 · 2021-06-25T18:20:17+0000

Answer:

a) In order to check if an estimator is unbiased we need to check this condition:

$E(\theta) = \mu$

And we can find the expected value of each estimator like this:

$E(\theta_1 ) = (1)/(7) E(X_1 +X_2 +... +X_7) = (1)/(7) [E(X_1) +E(X_2) +....+E(X_7)]= (1)/(7) 7\mu= \mu$

So then we conclude that
$\theta_1$ is unbiased.

For the second estimator we have this:

$E(\theta_2) = (1)/(2) [2E(X_1) -E(X_3) +E(X_5)]=(1)/(2) [2\mu -\mu +\mu] = (1)/(2) [2\mu]= \mu$

And then we conclude that
$\theta_2$ is unbiaed too.

b) For this case first we need to find the variance of each estimator:

$Var(\theta_1) = (1)/(49) (Var(X_1) +...+Var(X_7))= (1)/(49) (7\sigma^2) = (\sigma^2)/(7)$

And for the second estimator we have this:

$Var(\theta_2) = (1)/(4) (4\sigma^2 -\sigma^2 +\sigma^2)= (1)/(4) (4\sigma^2)= \sigma^2$

And the relative efficiency is given by:

$RE= (Var(\theta_1))/(Var(\theta_2))=((\sigma^2)/(7))/(\sigma^2)= (1)/(7)$

Explanation:

For this case we assume that we have a random sample given by:
X_1, X_2,....,X_7 and each
$X_i \sim N (\mu, \sigma)$

Part a

In order to check if an estimator is unbiased we need to check this condition: