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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:

  1. 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)
  2. DF is as unknown as $\mu$ and $\sigma$, so all 3 parameters need to be estimated.
  3. 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)?

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possible duplicate of Estimating parameters of Student's t-distribution –  Scortchi Feb 21 at 11:29
    
Sorry, that link's for known degrees of freedom. Still, see @Lucozade's answer - you need to maximize that likelihood numerically. With only three parameters it's easy to sense-check the answer with graphs & tweak starting values if you need to. R's optim should do it, with a choice of methods; & also gives the Hessian for working out the standard errors of your estimate. (BTW, EM doesn't require missing data.) –  Scortchi Feb 21 at 11:46
    
okay, so i should estimate reasonable df values and use something like profile likelihood to then solve (or graph and look for maximum). I'm looking at the EM answer on the thread you linked to and am a bit confused for the working. Do you sub 0 and df/df-2 for mu and sigma into the bottom wi equation and then continue? –  user40124 Feb 21 at 12:43
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I was suggesting forgetting EM & using any old method to maximize the likelihood function of all three parameters directly, checking graphically that the likelihood function is well-behaved. You can get at starting values by comparing kernel density estimates of the real data with simulated ones. (There are probably better ways, which I why I didn't put this as an answer, but brute force should work for this simple case.) –  Scortchi Feb 21 at 13:03

1 Answer 1

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.

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Actually, there are two conflicting definitions for the three parameter t distribution. One is the non-central t as you define here. The other is the location-scale transform of a standard t variate. Which does not even require EM. –  Xi'an Oct 29 at 19:29

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