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Note I am considering the standard $t$ distribution $(\mu=0,\sigma=1)$

The method of moments for $\nu>2$ is derived in this question

My question is, if the true (population) value of $\nu$ is $2$, and we don't know this, then what is the MOM estimator estimating?

My confusion is that in this case the MOM estimator uses higher order moments that don't exist, so does this mean that if $\nu=2$ the MOM estimator will never get this right?

More generally, if we use a MOM estimator based on moments that only exist for certain parameter values, what happens if the true parameter values are not in that range?

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    $\begingroup$ Even when $\nu=3$ or $4$, say, the MOM estimator may be quite poor at typical sample sizes. Just because a MOM exists doesn't mean it will necessarily be much use. Note that if the mean and variance are unknown, the estimation of $\mu,\sigma^2$ and $\nu$ could mean using up to 4th moments ... whose variance relies on 8th moments. If you have $\nu\leq 8$ your estimator for $\nu$ doesn't have finite variance. $\endgroup$ – Glen_b -Reinstate Monica Jun 26 '17 at 5:36
  • $\begingroup$ @Glen_b I see. What I am asking though is whether the MOM estimator will be able to (correctly) predict that $\nu=2$ (if that is the true value), asymptotically (i.e. if we have an infinitely large sample size). If the true $\nu>2$, then I believe the MOM is consistent for $\nu$, but we derived the MOM estimator assuming $\nu>2$, so if the true value of $\nu=2$, I'm not sure whether the consistency of the MOM estimator means anything, since we derived it using a moment that doesn't exist in the population then? $\endgroup$ – user106860 Jun 26 '17 at 17:10
  • $\begingroup$ It doesn't have anything to converge to (in that the function of moments has an undefined expectation). Horst already answered that in the first sentence of his answer. I was giving information to indicate it might not be of much practical use even when it converges to the population value. $\endgroup$ – Glen_b -Reinstate Monica Jun 26 '17 at 20:32
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In these cases, the estimators are no more consistent. The consistency of a working MOM estimator is guaranteed by the consistency of the moment estimators plus the fact that the parameter is a continuous function of the moments.

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