# Why is are unbiased statistics used more commonly than statistics with lower MSE?

I understand the difference between consistency and bias; one converges as the sample size increases, and the other converges as the number of estimates increases, respectively.

But, I don't understand why people generally prefer the unbiased version of sample variance to estimate population variance. Using simulation, it can be shown that a biased estimator of the population variance can have lower MSE than an unbiased estimator.

So, why do we use the unbiased version of sample variance if the MSE can be higher?

Another question might be how to accurately estimate the MSE without using simulation, since a big component of MSE is Variance of the estimator, and simulation is often not possible since it requires knowing the true value of the parameter/estimand (in this case, the estimand is the poulation variance).

http://www.johndcook.com/blog/bias_consistency/

## 1 Answer

This might sound like a pun, but it is really about bias - variance trade offs. Use the biased version, and you have less variance but bias. Use the unbiased one, and you have no bias but more variance.

I think another, perhaps less valid reason, is that many people - when the learn statistics, tend to think of unbiasedness as an (very) important property. Hence they pick this version.

I any case, if you wish to do inference you will most likely need to apply some form of asymptotic result - in this case both estimators should do the trick equally well (or bad, as some might put it). And the choice is less important.