Posts / Folded medians

In the previous post, we discussed the Gastwirth’s location estimator. In this post, we continue playing with different location estimators. To be more specific, we consider an approach called folded medians. Let $x = \{ x_1, x_2, \ldots, x_n \}$ be a random sample with order statistics $\{ x_{(1)}, x_{(2)}, \ldots, x_{(n)} \}$. We build a folded sample using the following form:

$$ \Bigg\{ \frac{x_{(1)}+x_{(n)}}{2}, \frac{x_{(2)}+x_{(n-1)}}{2}, \ldots, \Bigg\}. $$

If $n$ is odd, the middle sample element is folded with itself. The folding operation could be applied several times. Once folding is conducted, the median of the final folded sample is the folded median. A single folding operation gives us the Bickel-Hodges estimator.

In this post, we briefly check how this metric behaves in the case of the Normal and Cauchy distributions.

Simulation study

Let’s conduct the following simulation: TODO

Here are the results:

The observations:

The folded median approach could be practically interesting in some light-tailed cases because of its high efficiency.


References (1)