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The parameters of a t distribution can be estimated via

  1. maximum-likelihood estimation (ML) or
  2. method of moments (MM)

If we use the method of moments we have:

$$\mu=E(R)$$

$$\sigma^2=V(R)=\frac{\beta \nu}{\nu -2}$$

$$\kappa = \frac{6}{\nu-4}$$

we can rewrite the last two equations:

$$\beta = \frac{\sigma^2}{(\nu/(\nu-2))}$$ and $$\nu=\frac{6}{\kappa}+4$$

Now my question is, for MM, how do I estimate those parameters empirically? I mean, is it ok to do the following? :

  • use R and calculate mean(data), use this value as an estimate for $\mu$.
  • use R and calculate var(data) and kurtosis(data), insert these values into the equations and get the values $\beta$ and $\nu$.

Is this correct? Are the resulting values the same as using ML?

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    $\begingroup$ Write out the parameterization you are using, and define all the symbols. $\endgroup$
    – StasK
    Mar 19 '13 at 14:44
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    $\begingroup$ "Are the resulting values the same as using ML?" Quite plainly, no. $\endgroup$
    – Glen_b
    Mar 19 '13 at 14:51
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Of course the parameter estimates will be different from ML and MM. For one thing, the ML estimate is some sort of robust mean, with tails downweighted, as opposed to MM-based brute force arithmetic mean.

First of all, for the MM to work, you will need to have higher order moments to ensure that the sums necessary for the MM converge. E.g., for the sum of the fourth moments to be asymptotically normal, you need to have finite 8th order moments; for it to converge in the sense of the law of large numbers, you may need just another bit of a moment on top of the fourth, but at any rate, you have limitations on the degrees of freedom of your $t$ distribution. If you come to the data without knowing that your d.f. is >10, then you can find yourself in a situation where the MM estimator is not even consistent. High variability in the estimates, even in the situation when the estimator is defined and consistent, will also make MM less asymptotically efficient than ML.

ML does not have these theoretical problems, but in practice, it is found through numerical maximization, and in the particular case of $t$-distributions is prone to multiple local maxima. (By particular case, I mean that generally people speculate whether you have multiple local maxima or not, but in the case of $t$-distribution, existence of local maxima can be demonstrated analytically.) So you have to try a bunch of different starting values for it. Having a strong functional dependence between the variance and the shape parameter slows down convergence, as well.

As a bottom line, I don't see much use for MM... except probably for starting points for ML.

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    $\begingroup$ +1, especially for the initial observations about the need to assume finite higher moments. As a point of clarification I would like to suggest that we carefully distinguish an estimator from the procedures used to calculate it. Thus I would not say that ML is an "iterative estimator" (whatever that might mean, actually): it is only the case that numerical procedures to compute ML estimates typically are iterative. The presence of "multiple local maxima" can indeed be a problem: but only in practice, not in theory, provided there is a unique global maximum. $\endgroup$
    – whuber
    Mar 19 '13 at 16:37
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    $\begingroup$ @whuber, good comment -- see edit to address it. In econometrics, there are iterative generalized-moments estimators that use the converged results of the previous stage to initialize the optimization problem for the next one; their convergence properties are often dismal, as you can imagine. $\endgroup$
    – StasK
    Mar 20 '13 at 3:43
  • $\begingroup$ mh, this does not really help me much, since I am lost and don't know what to do now. The point is, I have financial data and want to fit a t distribution to use it for VaR calculation. E.g. this paper: ideas.repec.org/a/eme/jrfpps/v7y2006i3p292-300.html does the same as I wrote: They are just calculating the estimates and insert them, or am I wrong? So there is nothing about the "convergence of some sums", what I did not even understand, what sums? Anyway, a real empirical help/answer would help me a lot! @StasK $\endgroup$ Mar 20 '13 at 15:40
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    $\begingroup$ For platykurtic small sample data, mom will lead to negative dof for Student's t distribution. One needs to be careful while using these estimates. $\endgroup$
    – Kumar
    Apr 20 '16 at 14:55

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