#StackBounty: #maximum-entropy #fat-tails How to express tail index as an expectation (for MaxEnt procedure)

Bounty: 50

I’m trying to construct a prior probability density function, $f_X(x)$, for a fat-tailed distribution using the maximum entropy (MaxEnt) method. For my known “testable” information, I have the following information available:

$$
mathbb{E}[X^2] = sigma^2 \
text{Tail index} = alpha,
$$
where the tail index is defined as the exponent $alpha$ such that
$$
lim_{x to infty} (1 – F_X(x)) sim kx^{-alpha},
$$
where $F_X(x)$ is the CDF and $k$ is a proportionality constant.

The MaxEnt procedure requires the $n$ statements of testable information to be of the following form:
$$
mathbb E [f_i (X)] = F_i, qquad i = 1, ldots, n.
$$
It is clear in our case that $f_1(x) = x^2$ and $F_1 = sigma^2$.

What function $f_2(x)$ satisfies $mathbb E[f_2(x)] = alpha$? Is it possible to formulate the problem in this manner?


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