# mle estimate for standard deviation of t-distribution

For a Student-t distribution, $t_{\nu}\left(\mu,s^2\right)$, let $\hat{s}$ be mle of scale and $\hat{\nu}$ be the mle of degrees of freedom. Functional invariance of mle implies that any linear or non-linear estimators of mle are also mle. Hence, $\sqrt{\frac{\hat{\nu}-2}{\hat{\nu}}}$ is also mle. What can I say about the product of 2 mles. Is $\hat{\sigma}=\hat{s}\sqrt{\frac{\hat{\nu}-2}{\hat{\nu}}}$ also an mle estimator?

Can someone give a complete statement for functional invariance of mle with reference?

• Unless you take a very narrow view of "functional invariance," you have already answered this question. What, then, do you mean by this phrase? And is your "product of 2 mles" a product of two estimates (via Maximum Likelihood) from the same or different datasets? – whuber Feb 20 '15 at 19:52
• Wiki for invariance of mle says , "If the parameter consists of a number of components, then we define their separate maximum likelihood estimators as the corresponding components of the MLE of the complete parameter". So sum and products of mle must be mle and $\hat{\sigma}$ should be mle. – Kumar Feb 20 '15 at 20:03
• Similarly, $\hat{\mu} + \hat{\sigma}t_{\nu}^{-1}\left(0,1\right)$ must be an mle for the quantile – Kumar Feb 20 '15 at 20:12

From this point of view the question does not exist! Estimated parameters, such as $\hat\nu$ and $\hat s^2$, are coordinates of the lowest point. Therefore functions of the estimates, such as $\hat {s}\sqrt{(\hat\nu - 2)/\hat\nu}$, are well-defined properties and will not change if you change the parametrization, just as the actual latitude and longitude of a point will not change just because you choose to use UTM coordinates to say where that point is. (Obviously you would need to take care not to confuse the UTM easting with $\hat\nu$ and the UTM northing with $\hat s^2$: that would be quite an error, wouldn't it?)