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Given yield measurements $X_1,X_2,X_3$ from three independent runs of an experiment with variance $\sigma^2$, which is the better of the two estimators: $\hat\theta_{1}$= $\frac{X_1+X_2+X_3}{3}$, $\hat\theta_{2}$=$\frac{X_1+2X_2+X_3}{4}$

I know that in order to find the best estimator if both are unbiased, we are supposed to choose the one with the smallest variance. I need help just starting this problem. Thank you.

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Variance of the Sample mean estimate is just $$\text{VAR}(\bar{\mu})=\text{VAR}\left(\frac{\sum_i X_i}{n}\right)$$ for the first case and $$\text{VAR}(\bar{\mu_2})=\text{VAR}\left(\frac{X_1+X_2+X_2+X_3}{4}\right)$$ in the second. Now apply scaling property of variances to see which one has a higher variance.

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Hints:

  • $\text{Var}(X_1+X_2) = \text{Var}(X_1)+\text{Var}(X_2)$ if they are independent

  • $\text{Var}(k X) = k^2 \text{Var}(X)$ and so $\text{Var}\left(\dfrac{X}{k}\right) = \dfrac{\text{Var}(X)}{k^2} $

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