How to understand the Repeated Median Estimator (Siegel 1982)?

Question

I guess I did not understand the formula of the repeated median estimator introduced by Siegel (1982):

This is the formula:

If I got the following time series for instance:

y = { 2.3  , 5.6 , 7.8,  100, 110, 8, 4 }

Clearly $i$ and $j$ run from 1 to 7 as $|y|=7$. But how can I compute $\hat{\beta_0}$? Thanks for your help.

Answer 1

Just consider starting with 2.3 and calculating the Slope to all other values in the dataset. Then taking the Median Slope and associating it with the 2.3 value. Then move on to 5.6 and repeat the process. At the end of this you will have a Median Slope for each value. Then you extract the median of these Median Slopes. This is why it is called a "Repeated" Median Estimator.