Mean Squared Error

Can be used to find the quality of an Estimator. A popular Loss Function to measure how wrong a model’s predictions are.

Formula

$MSE=Var+Bia{s}^2$

Definition

• For $\theta$ as an unknown parameter
• Suppose $T$ is an Estimator of $\theta$
• $MS{E}_{\theta }(T)=E_{\theta }[(T-\theta {)}^2]$

Alternate Definition

• MSE As Dart Boards

MSE Proof

• $MSE[T]=var[T]+(E[T]-\theta {)}^2$
• Note that $V[T-\theta ]=E[(T-\theta {)}^2]-E[T-\theta {]}^2$
• $⟹V[T]=MSE[T]-(E[T]-E[\theta ]{)}^2$
• $⟹V[T]=MSE[T]-(E[T]-\theta {)}^2$
• $⟹V[T]=MSE[T]-(Bias(T){)}^2$
• $⟹MSE=V[T]+Bias[T{]}^2$