# #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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