Resistance to the low-density regions: the Harrell-Davis median

In the previous post, we defined the resistance function that show sensitivity of the given estimator to the low-density regions. We also showed the resistance function plots for the mean and the sample median. In this post, we explore corresponding plots for the Harrell-Davis median.

The resistance function

As was shown in the previous post, we define the function of resistance to the low-density regions as follows:

\[R(T, n, s) = \max_{s \leq k \leq n} R(T, n, s, k), \]

\[R(T, n, s, k) = |T(\mathbf{x}_k) - T(\mathbf{x}_{k-s})|, \]

\[\mathbf{x}_k = \{ \underbrace{0, 0, \ldots, 0}_{k}, \underbrace{1, 1, \ldots, 1}_{n-k} \}, \]

where \(T\) is an estimator, \(n\) is the sample size, \(s\) is the number of sample values that jump from the first mode to the second one.

Resistance of the Harrell-Davis median

Now it’s time to build the plot of \(R(T, n, s)\) that compares the mean, the sample median, and the Harrell-Davis median. In this experiment, we consider \(n \leq 100\), \(s \in \{1, 2, 3, 4, 5, 6\}\). Here are the plots:

As we can see, the Harrell-Davis median is not only more statistically efficient to low-density regions, but it is also more resistant to the low-density regions.

In future posts, we will explore the resistance function for other measures of central tendency.