Question

Suppose that $\hat{\theta_1}$, $\hat{\theta_2}$ are unbiased estimators of $\theta$ with $Var(\hat{\theta_1}) = 1$ $Var(\hat{\theta_2}) = 1$ $Cov(\hat{\theta_1}, \hat{\theta_2}) = \frac{1}{3}$ Define another estimator, $\hat{\theta_3}$, by $\hat{\theta_3} = \frac{1}{2}\hat{\theta_1} + \frac{1}{3}\hat{\theta_2}$ a) $\hat{\theta_3}$ is a biased estimator of $\theta$. Multiplying $\hat{\theta_3}$ by what number will make it unbiased? Express your answer as a fraction. b) What is the variance of the estimator $\hat{\theta_3}$? Make sure to take into account the covariance term.

          Suppose that $\hat{\theta_1}$, $\hat{\theta_2}$ are unbiased estimators of $\theta$ with
$Var(\hat{\theta_1}) = 1$
$Var(\hat{\theta_2}) = 1$
$Cov(\hat{\theta_1}, \hat{\theta_2}) = \frac{1}{3}$
Define another estimator, $\hat{\theta_3}$, by
$\hat{\theta_3} = \frac{1}{2}\hat{\theta_1} + \frac{1}{3}\hat{\theta_2}$
a)
$\hat{\theta_3}$ is a biased estimator of $\theta$. Multiplying $\hat{\theta_3}$ by what number will make it unbiased? Express your answer as a fraction.
b)
What is the variance of the estimator $\hat{\theta_3}$? Make sure to take into account the covariance term.

$Suppose that θ̂1̂, θ̂2̂ are unbiased estimators of θ with Var(θ̂1̂) = 1 Var(θ̂2̂) = 1 Cov(θ̂1̂, θ̂2̂) = (1)/(3) Define another estimator, θ̂3̂, by θ̂3̂ = (1)/(2)θ̂1̂ + (1)/(3)θ̂2̂ a) θ̂3̂ is a biased estimator of θ. Multiplying θ̂3̂ by what number will make it unbiased? Express your answer as a fraction. b) What is the variance of the estimator θ̂3̂? Make sure to take into account the covariance term.$

Added by Miriam S.

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