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That is, broadly speaking, as the tail gets heavier, there's a tendency for (at a fixed value of $\sigma^2$) the distribution to get "peakier" at the same time (more kurtotic, in a loose sense). This is not, however, a certain thing - it tends to be the case across a broad range of commonly considered densities, but it doesn't always hold. When it does hold, the variance of the median will reduce (because the distribution has more probability in the immediate neighborhood of the median), while the variance of the mean is held constant (because we fixed $\sigma^2$).

If we had instead used examples from say the t-distributions, broadly similar effects would be seen, but the progression would be different; the crossover point is a little below $\nu=5$ df (actually around 4.68) -- for smaller df the median is more efficient, for large df the mean is.

That is, broadly speaking, as the tail gets heavier, there's a tendency for (at a fixed value of $\sigma^2$) the distribution to get "peakier" at the same time (more kurtotic, in Pearson's original, if loose, sense). This is not, however, a certain thing - it tends to be the case across a broad range of commonly considered densities, but it doesn't always hold. When it does hold, the variance of the median will reduce (because the distribution has more probability in the immediate neighborhood of the median), while the variance of the mean is held constant (because we fixed $\sigma^2$).

If we had instead used examples from say the t-distributions, broadly similar effects would be seen, but the progression would be different; the crossover point is a little below $\nu=5$ df (actually around 4.68) -- for smaller df the median is more efficient (asymptotically), for large df the mean is.

If we make the distribution peakier still for a given variance, (perhaps by making the tail heavier than exponential), the median can be far more efficient (relatively speaking) still.

If we make the distribution peakier still for a given variance, (perhaps by making the tail heavier than exponential), the median can be far more efficient (relatively speaking) still. There's really no limit to how high that peak can go.

The relative variance depends on sample size. It's common to discuss the ratio of ($n$ times the) the asymptotic variances, but we should keep in mind that at smaller sample sizes the situation will be somewhat different. (The median often does a bit better than its asymptotic behaviour would suggest. For example, at the normal with $n=3$ it has an efficiency of about 74% rather than 63%. The asymptotic behavior is generally a good guide at quite moderate sample sizes, though.)

First, the hyperbolic secant ($\text{sech}$) distribution, for which the standard form has variance 1 and height at the center of $\frac{1}{2}$, so the ratio of asymptotic variances is 1 (the two are equally efficient in large samples).

At finite sample sizes, it's sometimes possible to compute the variance of the distribution of the median explicitly. Where that's not feasible - or even just inconvenient - we can use simulation to compute the variance of the median (or the ratio of the variance*) across random samples drawn from the distribution (which is what I did to get the 74% figure above).

The relative variance depends on sample size. It's common to discuss the ratio of ($n$ times the) the asymptotic variances, but we should keep in mind that at smaller sample sizes the situation will be somewhat different. (The median sometimes does noticeably better or worse than its asymptotic behaviour would suggest. For example, at the normal with $n=3$ it has an efficiency of about 74% rather than 63%. The asymptotic behavior is generally a good guide at quite moderate sample sizes, though.)

First, the hyperbolic secant ($\text{sech}$) distribution, for which the standard form has variance 1 and height at the center of $\frac{1}{2}$, so the ratio of asymptotic variances is 1 (the two are equally efficient in large samples). However, in small samples the mean is more efficient (its variance is about 95% of that for the median when $n=5$, for example).

At finite sample sizes, it's sometimes possible to compute the variance of the distribution of the median explicitly. Where that's not feasible - or even just inconvenient - we can use simulation to compute the variance of the median (or the ratio of the variance*) across random samples drawn from the distribution (which is what I did to get the small sample figures above).