Definition of Tail-index of a probability distribution

Question

What is a valid definition of Tail-index of a probability distribution?

I understand that it is something to do with the rate of convergence of the density function $f(x)$ $($to $0)$ as $x \to \infty$. I tried searching google and I do find a lot of articles/papers on the topic but nowhere I can find a specific definition of the term.

Any help would be much appreciated! Thanks.

Answer 1

I have found a defintion at http://freakonometrics.hypotheses.org/2338, although again it is not presented as a primary definition of "tail index" but rather as an adjunct to the discussion of what they call heavy-tailed distributions. So I have had to fill in some parts to add a bit of rigor, and other people should check these parts.

The parts obtained from the freakonometrics article are in the shaded areas.

Consider any distribution $P(X)$ with cumulative distribution function $F(x) = 1- \overline{F}(x)$ defined by $\mbox{Pr }(X > x) = \overline{F}(x)$, such that for some $\xi>0$, $$ \overline{F}(x) = x^{-1/\xi}\mathcal{L}(x) $$ where $\mathcal{L}(x)$ is some slowly varying function for large $x$.

The tail index of the fat-tailed distribution $P(X)$ is by definition $\xi$.

Although the freakomomics article calls this a heavy-tailed distribution, in Wikipedia and elsewhere this is called a fat-tailed distribution. The definition of a "slowly varying function" is that for all $a>0$ $$ \lim_{x\to\infty}\frac{\mathcal{L}(ax)}{\mathcal{L}(x)}=1 $$ Thus we can restate the condition as $\overline{F}(x) \sim x^{-1/\xi}$ for some $\xi>0$, where $\sim$ denotes asymptotic equivalence.

The definition of a heavy-tailed distribution given in https://en.wikipedia.org/wiki/Heavy-tailed_distribution is that $P(X)$ is a heavy-tailed distribution if for all $\lambda > 0$, $$ \lim_{x\to\infty}e^{\lambda x}\overline{F}(x) = \infty $$ All fat-tailed distributions are heavy-tailed in this sense, but not vice-versa.

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Added example in response to comments

For example, consider the probability distribution function $$f(x)=\left\{\matrix{0&x<1\\\frac{e^{1-\sqrt{x}}}{2\sqrt{x}}&x\geq 1}\right. \\\overline{F}(x)= e^{1-\sqrt{x}}\mbox{ for } x\geq 1 $$ For any positive $\lambda$ $$\lim_{x\to\infty}e^{\lambda x}\overline{F}(x) = \infty$$ so the distribution is heavy-tailed. But for any positive $\xi$, $$ \lim_{x\to\infty}x^{1/\xi}\overline{F}(x) = 0 $$ which implies that the distribution is not fat-tailed

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Equivalently, there exists a slowly varying function $\mathcal{L}^*(x)$ such that for $0<p<1$, $$ \log F^{-1}(1-p) = - \xi \log p + \log\mathcal{L}^*(1/p)$$

$\xi$ can by visualized as the opposite [negative] of the slope, at small $p$, of $\log F^{-1}(1-p)$ when that is plotted against $p$.

Somebody should add this definition to the Wikipedia page on heavy-tailed distributions, just above the section on Pickand's estimator of the tail index. However, I think the Freakonometrics reference is inadequate, both because it is not primarily intended as a definition of the term, and because the confusion about heavy-tailed and fat-tailed reduces confidence in using that as a reference.

Answer 2

For an academic reference, a definition is given in the paper "The impact of competition on prices with numerous firms" by Xavier Gabaix, David Laibson, Deyuan Li, Hongyi Li, Sidney Resnick and Casper G. de Vries (Journal of Economic Theory, 2016) -- see Definition 1. For any distribution with cdf $F$ and pdf $f$, and support $(w_{l}, w_{u})$, the tail index $\gamma$ is defined

$$ \gamma := \lim_{x \to w_{u}}\frac{d}{dx}\left(\frac{1-F(x)}{f(x)}\right).$$

They provide a number of examples in their Table 1.