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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?

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  • $\begingroup$ 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 $\endgroup$ – P. Quinton May 6 '19 at 6:00
  • $\begingroup$ It is usually defined as $E\,\lvert\rvert \hat {\boldsymbol b}-\boldsymbol b\lvert\rvert^2$, consistent with the risk function for quadratic loss. $\endgroup$ – StubbornAtom May 7 at 21:47
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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. $$ \varepsilon = {{\left| {\Delta {\bf v}} \right|} \over {\left| {{\bf v}_{\,ref} } \right|}} $$

Error_Vector_1

That means that you are considering the error as the distance between the vectors' tips.

In certain situations (when the focus is on energy) it may be of interest to distinguish between the normal (out-of-phase) and the parallel (in-phase) component of the error.

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