t-distribution parameter estimation I know there are already several threads on this, but none seem to explicitly cover what I want. I have a set of financial data (pulled straight from Bloomberg) and am trying to fit a t-distribution (with theoretical discussion of MLE).
I know it can't be solved in closed form and am looking at EM and Newton-Raphson currently, but some issues have come up:


*

*I'm not sure there's any way I can justify that my data is incomplete (data is closing price of stock index over last year)

*DF is as unknown as $\mu$ and $\sigma$, so all 3 parameters need to be estimated.

*Data is univariate


The above methods seem to need known df or incomplete data. Is this strictly true? Do I need to modify the methods (if so please show) or choose new ones (if so please specify)?
 A: It sounds like you are taking too narrow a view of incomplete data in the context of the EM algorithm. Latent variables may indeed be unobservable due to issues with the measurement process but they can also correspond to more abstract concepts.
The $t$-distribution admits the following hierarchical decomposition:
$$
y_i \sim t(\mu,\sigma^2,\nu)\\
y_i|\tau_i \sim N\left(\mu,\frac{\sigma^2}{\tau_i}\right)\\
\tau_i \sim Gamma\left(\frac{\nu}{2},\frac{\nu}{2}\right)
$$
Under this setting, the latent variables are the $\tau_i$. The idea with the EM algorithm is that if you had observed the $\tau_i$ then maximization of the complete log-likelihood would be straightforward.
In fact, I do not think the EM algorithm gains much in MLE of the $t$-distribution as conditional maximization of the complete log-likelihood at the M step for $\nu$ can still only be done numerically. True, it helps that this optimization is one dimensional but the EM algorithm will most likely be slower to converge than direct three dimensional maximization of the likelihood.
