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what is a medcouple? i understand that it is the median of a couple of data points but it is not clear to me what these pairs of data actually are. e.g. https://wis.kuleuven.be/stat/robust/papers/2008/adjboxplot-revision.pdf this paper explains it. i am struggling to understand what this kernel is doing.

can somebody explain?

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Im sorry, im not very good with formulars/formating but i still try my best to share my understanding of med couple.

We have the MC itself as MC = med h(xi, xj)

and we have h as h(xi, xj) = (xj−Q2)−(Q2−xi) / (xj−xi)

we have two indices: i and j. they are used to form the couples to be compared. The goal is to compare the biggest value in the data-set to the smallest one. Then compare the second-biggest to the seccond smallest The 3rd biggest to the 3rd smallest and so on.

This can be done by sorting the dataset descending with j starting at 1 and i starting at [length of data] With every step j gets increased by one and i decreased by 1.

example-dataset, sorted descending allread: {10, 8, 5, 2, 1}

x[j=1] will start on the left side and select 10. x[i=5] will start on the right side and select 1. Thats the first couple for the medcouple calculation

For step 2 j gets increased by 1 and i decreased.

x[j=2] will start on the left side and select 8. x[i=4] will start on the right side and select 2. Thats the second couple for the medcouple calculation (and also the final one in our example, since there are no pairs left and right the median left)

Now we can plug this couples into the h(xi, xj) = (xj−Q2)−(Q2−xi) / (xj−xi) formular. (xj−Q2) is a measure of how much the bigger of the 2 couple-values lies above the median (Q2−xi) is a measure of how much the smaller of the 2 couple_values lies under the median. you could also see this as distance to the median for both of the couple values.

(xj−Q2) - (Q2−xi) evaluates the difference in distances of the couple-values to the mean. If this value is negative you know, that the smaller value of the couple lies further away from the median than the bigger value of the couple.

By dividing (xj−Q2) - (Q2−xi) by (xj−xi) you standardise the difference of the couple-values distance to the median by their distance to each other.

For each of your value couples (from step1 to stepn, see above) you now have value, that tells you which of this two values lies further away from the median and how impactfull this difference is in relation to the distance between both values from the couple (remember: wenn this value is < 0 it means that the lower value is further away from the median, than the higher value).

Now we use the median over all this calculated values, to get a measurement for the skewness of the distribution. If we find, that this calculated med lies below 0 we know, that there are more couples where the smaller value is further away from the median than the larger value, or in other words, the left-side tail of the distribution is longer than the right-side one.

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