# When would maximum likelihood estimates equal least squares estimates?

Under what situations would MLE (Maximum Likelihood Estimates) equal LSE (Least Squares Estimates)?

I got an impression that under norm 2 ($L_{2}$), MLE and LSE are equal.

For example, the process of solving $\mathrm{min}||y−Ax||_{2}$ is actually the MLE estimation of parameter $A$ for random variable $y=Ax+\epsilon$ where $x$ and $\epsilon$ is normal.

However, is that generally true that the minimization problem under $L_{2}$ norm is the same as maximum likelihood estimation? For example, consider a quadratic function $f(X) = XAX$, can minimizing the distance between $f(X)$ and some value $Y$ under $L_{2}$ can be solved by MLE?

• Well take $\varepsilon$ to come from other distribution than normal. Take the distribution which does not have second moments, then the expectation of the $L_2$ norm need not exists. This is a contrived example, but it might give general idea, i.e. that except in normal case MSE and LSE are usually not equal. – mpiktas Jul 5 '13 at 7:45
• Also note that even in the Gaussian case OLS and MLE are only equivalent for estimating $\mu$ (and decomposition thereof). For $\sigma^2$ they are not equivalent. – Momo Jul 5 '13 at 8:48
• In addition to mpiktas' and Momo's excellent comments, see this reference on page 44. The MLE for $\sigma^{2}$ is biased. – COOLSerdash Jul 5 '13 at 12:13