Variance estimate for Student's t-distribution with heavy tails The variance of a Student's t random variable when the degrees of freedom $\nu$ is greater than $2$ is $\nu/(\nu-2)$. In R, when I try to estimate the variance using the usual estimator $$1/(n-1)\sum_{i=1}^n (x_i-\overline{x})^2$$, the estimates seem biased, when $\nu$ is close-ish to 2, like $\nu=2.2$. Here is my code and the output:
nu <- 2.2
sim <- sapply(round(seq(1e6, 5e6, len=10)), 
        function(n)replicate(10, var(rt(n, 
          nu)) - nu/(nu-2)))
matplot(t(sim), pch=1, col=1)
abline(h=0)


The estimated variance looks too small compared to the true variance. Is this

*

*my sample sizes (up to 5 million) are still too small for the consistency to show for degrees of freedom this close to 2,


*the sample variance estimator actually isn't consistent for these values of $\nu$,


*something else?

Your example would be a good application of a new estimator for heavy-tailed distributions.  I call the method Independent Approximates (IAs), since it utilizes a subsample of n-tuples that are approximately equal.
The method can be applied to estimate the location $\mu$, scale $\sigma$, and degree of freedom $\nu$. Since you are estimating the variance, I'll assume the scale is your primary interest.
To estimate the scale, assuming the location is known, one can use the triplet IAs. The triplet IAs are selected by partitioning the original samples into triplets and subselecting those triplets that are approximately equal, and retaining the median sample.  The referenced paper provides further details.  The triplet-IAs which are guaranteed to have a finite second moment for all $\nu$, and if $\nu > 2$ the variance of the second moment will be finite. For your example with $\nu=2.2$, the median triplet IA samples will have $\nu_{triplet} = 8.6$.
The estimate of the second moment of the triplets, $\mu_{triplet}^2$ can be used to estimate the scale of the original distribution using the following function $\mu = \sqrt{3 \mu_{triplet}^2}$.
See my paper Independent Approximates enable closed-form estimation of heavy-tailed distributions for further details and examples.
