I am looking for a theoretical expression for the ARE of mean and median for student-t distribution against sample size (degree of freedom), exactly the blue curve shown in John Cook's blog. Can anyone point me to the right reference with the expression for it?
Thanks
Forewarning: The following ARE expression is wrong, the simulation result at the end is more correct
Following steps from this paper reference
For an arbitrary distribution $\mathcal{F}$ with probability density $f(x)$ Asymptoptic relative efficiency (ARE) between median function and mean function for estimating the central parameter $\theta$ is: $4[f(\theta)]^2\sigma^2_F$ (from 1).
The ARE for a Student-t distribution is then found by substituting $f(\theta=0)$ of a Student-t distribution which yields:
$\operatorname{ARE}=\frac{4\nu}{\nu-2}\left\{\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\nu\pi}}\right\}^2$ for $\nu>2$
However, due to the expression made for the sample distribution of a median function is only valid for large sample size (again refer to 1), the ARE expression above does not really reflect reality. In fact, an interesting behavior is found in simulations.
Using MATLAB software and generating a Gaussian distributed random process with variance of unity and zero mean is sampled $N$ times to generate $N$ samples. The mean function and median function is used to estimate the location (central tendancy) of these samples. We are in effect estimating the population mean of the Gaussian population, which is zero. Theory dictates the sampling produces Student's $t$-distribution with degrees of freedom equal to $N-1$. This is repeated 10 million times and the variances are computed for both mean and median functions, their ratio is the Relative Statistical Efficiency. This experiment is performed for $N$ from between 2 to 16 inclusive.
The result is plotted below:
One can observe the zig-zag trend showing a dependence on whether sample size is even or odd. This is because the median is calculated differently depending if the sample size is even or odd: pairwise mean is calculated at the last step for even sample sizes. This is also why when sample size is two, the RSE is exactly one, as pairwise mean and median are essentially the same mathematical operation. It is interesting to see even sample size result in a higher efficiency for the median estimator.
The $\pi/2$ line is ths asymptote representing the ARE (for $N=\infty$) as noted in 1.
1 R. Serfling, 'Asymptotic relative efficiency in estimation', in International Encyclopedia of Statistical Science, M. Lovric, Ed., vol. 1, London: Springer, 2011, pp. 68–72.
