Highest Density Interval for the measure of central tendency

Question

When samples are skewed, mean is not a good estimation of central tendency. But instead, median is a better choice.

For cauchy distribution, I heard that there's a completely different estimator (which i didn't understand) that captures the central tendency of it.

And then I came to this new technique called the Highest Density Interval (HDI). It captures the range in which the distribution has the most amount of data points.

Question: If the goal is to measure the central location of a distribution, isn't it simpler to just always report HDI instead of worrying about mean, median, mode, and other more advanced estimators.

Answer 1

Statistical inference problems involving estimation of any aspects of a distribution arise only when the distribution is unknown. If the distribution is known then the centre can be located exactly, as can any other aspect of the distribution. Thus, even if you are interested in a HDI on the distribution rather than the mean, this is still something that would need to be estimated for an unknown distribution.

Now, the HDI is essentially just a generalisation of the mode (it shrinks down to the mode as its coverage probability approaches zero). It is certainly a useful object of inference in some cases, but it is not usually considered to be a measure of central location; indeed, in some cases the HDI can be quite far from the "centre" of the distribution (particularly when dealing with bimodal distributions). Whether estimation of the HDI is more important than estimation of the mean, median, etc., will depend on the context of the problem and what you want to know.