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?

  • $\begingroup$ 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? $\endgroup$
    – whuber
    Feb 20, 2015 at 19:52
  • $\begingroup$ 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. $\endgroup$
    – Kumar
    Feb 20, 2015 at 20:03
  • $\begingroup$ Similarly, $\hat{\mu} + \hat{\sigma}t_{\nu}^{-1}\left(0,1\right)$ must be an mle for the quantile $\endgroup$
    – Kumar
    Feb 20, 2015 at 20:12

1 Answer 1


I would like to share a mathematically accurate analogy with you. Maximum Likelihood estimation is like finding the lowest point on the earth's surface. The surface forms a manifold and the elevation is a function defined on that manifold.

We all know of many ways to describe points on the earth: we can use latitude and longitude, UTM coordinates, various national grid coordinates (elsewhere), 3D geocentric coordinates (which are used internally by GPS receivers), and far more. These descriptions are in terms of tuples of numbers--usually ordered pairs, but sometimes more (as in the 3D geocentric coordinates).

Does the lowest point depend on which coordinates you use? Of course not. The manifold and the elevation function are the mathematical reality, while the coordinates are (somewhat) arbitrary ways to help us do our calculations.

A set of probability models for your data also forms a manifold. The likelihood function acts just like the elevation. The maximum likelihood estimate is the set of all points with lowest elevation (usually, we hope, just one). How you describe them is up to you. The description is always in terms of numbers, called parameters.

Parameters can be considered "properties" of points, even if they might be somewhat arbitrary. For instance, latitude is a "property" of a point on the earth. It's fairly natural and might not be thought of as arbitrary: it tells us something real about that point. But longitude is arbitrary: it reflects the relationship between the point and a conventionally-established Prime Meridian. The proof of this arbitrariness is the existence of multiple different Prime Meridians (such as the Paris meridian). Regardless, because this convention is well-defined, we may equally well consider longitude to be a property of points.

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?)


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