Anatoly Vitold Stankyavichyus

This site is dedicated to the theory, computation, and applications of varentropy.
Varentropy is the variance of surprisal: it measures how much the information content −log f(X) changes from one observation to another. This makes it a natural statistic for comparing distributional shape, concentration, and rare-outcome severity. It is often more practical than tail-index estimation, because it avoids delicate tail-threshold choices, and it can remain meaningful when ordinary moments are unstable or infinite. With geometric assumptions, varentropy also gives testable structure: in one dimension, every log-concave distribution has varentropy at most 1, so a well-estimated value far above 1 is evidence against log-concavity.

You might be asking what's wrong with good old variance as a measure of risk? For starters, variance need not exist for fat-tailed distributions, because it needs a finite second moment.

Measuring and understanding the fat-tailedness of a process, or its distributional shape, matters for understanding how much of the risk/loss comes from a few rare observations under a given generating process. Under a subexponential law, the bulk of a large loss typically comes from a handful of extreme observations (the catastrophe principle), and the more pronounced this concentration, the less a finite sample reveals: for a well-behaved shape such as a Gaussian, even a small sample pins down the likely range of losses, whereas for an α-stable law with small α one should assume the most severe outcomes have yet to be seen.

In terms of Lévy processes, it's the difference between losses coming from the Brownian-motion part and losses coming from a jump. Variance only tells us the scale of losses, not where they come from.

In a nutshell, the importance of shape comes from the concentration of losses, and varentropy — together with mild geometric assumptions such as unimodality and monotone tails — is well suited to measuring it.