The mean squared error is a property of an [[estimator]] and can be used to compare two estimators to determine the "best" based on bias and variance. Let $\hat \theta$ be an [[estimator]] of a [[parameter]] $\theta$. The mean squared error (MSE) of $\hat \theta$ is defined as $MSE(\hat \theta) = E[(\hat \theta - \theta)^2]$ If $\hat \theta$ is an unbiased estimator of $\theta$, its mean squared error is simply the [[variance]] of $\theta$. The computation formula for MSE is $MSE(\hat \theta) = Var[\hat \theta] + (B[\hat \theta])^2$ where the variance of $\hat \theta$ is given by $Var[\hat \theta] = E[(\hat \theta - E[\hat \theta])^2]$ where the bias of $\hat \theta$ is given by $B(\hat \theta) = E[\hat \theta] - \theta$ ## relative efficiency Between two unbiased estimators $\hat \theta_1$ and $\hat \theta_2$, $\hat \theta_1$ is more efficient that $\hat \theta_2$ if $Var[\hat \theta_1] < Var[\hat \theta_2]$. Relative efficiency is a measure of how much more efficient $\hat \theta_1$ is. Relative efficiency is defined as $Eff(\hat \theta_1, \hat \theta_2) = \frac{Var[\hat \theta_2]}{Var[\hat \theta_1]}$ > [!Tip]- Additional Resources > - https://www.youtube.com/watch?v=XqWfeND04vs