# Mean squared error for vectors

I know that when we compare estimators $$\hat{b_1}$$ and $$\hat{b_2}$$ to an unknown parameter $$\beta$$, in classical statistics an estimator $$\hat{b_1}$$ is said to be "better" than $$\hat{b_2}$$ if:

$$MSE(\hat{b_1}) \leq MSE(\hat{b_2})$$ where MSE is the mean squared error: $$MSE(\hat{b_1}) = E((\hat{b_1}-\beta)^2 )$$

Now if I had a vector $$\boldsymbol{b} =(b_1,b_2,\ldots b_n)$$ of parameters to estimate, how could I compare estimators in terms of the MSE? Because there is no unique ordering relation in vectors.

I know some people compare component by component of both estimators, yet I seem to find no bibliography for that. Could you guys help me figure out a bibliography for that?

• You can join them in one vector and use your other formula, this is very standard but you should be careful about the fact that having a lot of parameters to estimate is really bad in terms of how much data you need to estimate them correctly – P. Quinton May 6 '19 at 6:00
• It is usually defined as $E\,\lvert\rvert \hat {\boldsymbol b}-\boldsymbol b\lvert\rvert^2$, consistent with the risk function for quadratic loss. – StubbornAtom May 7 at 21:47

In practical applications (engineering), the error (noise) vector is given by the modulus of the difference, in case taken relatively to the modulus of the reference vector, I.e. $$\varepsilon = {{\left| {\Delta {\bf v}} \right|} \over {\left| {{\bf v}_{\,ref} } \right|}}$$