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I do get that these concepts are well-defined, but I am referring to extreme value theory in particular, and the fact that these ideas are often tossed around as synonyms, leaving the details and subtleties restricted to people with very advanced knowledge of statistics or financial mathematics, while the rest pretend to know what we are talking about.

As an example, here is this Tweet by Paul Krugman with this intriguing interchange:

enter image description here

What does he mean by "when the Pareto coeff changes this is what u see."? Wouldn't it be more like, once it is a Pareto, it is fractal? Or if the alpha parameter is positive it is $\small \text{Pareto} \iff \text{Fréchet} \iff \text{fractal} \iff \text{power law} \iff \text{fat tailed}$?


QUESTION:

Is it true that $\small \text{Pareto} \iff \text{Fréchet} \iff \text{fractal} \iff \text{power law} \iff \text{fat tailed}$?

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  • $\begingroup$ Fractal means self similar, which in turn is associated with power laws, as fractals have a sort of characteristic exponent of self similarity (en.wikipedia.org/wiki/Hausdorff_dimension). However, not every power law necessarily has something to do with fractals. $\endgroup$ Jul 22 at 19:28
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Probably, the statement $\small \text{Pareto} \iff \text{Fréchet} \iff \text{fractal} \iff \text{power law} \iff \text{fat tailed}$ may not be completely accurate. For instance, there are distributions other than the Pareto with fat tails, and as pointed out in a comment, not all fat-tailed distributions exhibit fractal properties.

Power law distribution:

A random variable $X$ with domain $x \in \mathbb R^+$ follows a power law distribution if

$$\Pr[X ≥ x] = \bar F_X(x) ∼ cx^{−α}$$

with $c > 0$ and $α > 0.$

A power law decay exhibits scale invariance: $ f(kx)=c(kx)^{-\alpha}=k^{-\alpha}f(x)\propto f(x)$.

Pareto distribution:

The Pareto distribution is a power-law probability distribution, and makes reference to the type I with survival function

$$\bar F_X(x) = \begin{cases} \left(\frac{x_m}{x}\right)^{\alpha} & x> x_m \\ 1 & x< x_m \end{cases},$$

for some $α > 0$ and $x_m > 0.$ Alpha is the tail coefficient.

From here: "The Pareto distribution is a fractal probability distribution that has a power-law form with two important scaling (self-similar) properties: scaling under lower truncation and asymptotic scaling under addition."

Fractality:

From the same source: "Adoption of the Pareto distribution was independent of the study of fractals; however, the Pareto distribution is the probability distribution uniquely characteristic of fractals."

The scale invariance principle defines fractality and it is seen in power laws, and by extension in the Pareto distribution.

Fréchet distribution:

The idea is to get a sort of CLT for the maxima through two values: a 'mean' $b_n,$ and the equivalent of the standard deviation, $a_n>0,$ such that

$$\max(X_1,X_2,\cdots,X_n) = a_nX + b_n$$

which would be called a max-stable distribution. What it says is that distribution of the maximum is a re-scaling of the starting distribution (closure under maximization).

This is achieved with the Fisher-Tippet theorem (leads to the generalized extreme value distribution or GEV). The FT theorem results from the block maxima approach, i.e. the data is partitioned, and the maximal values in each partition follow a GEV of the form

$$F(x;\mu.\sigma;\xi)=e^{-(1+\xi \frac{x-\mu}\sigma)^{-1/\xi}}$$

Fisher-Tippet theorem: Let $X_1,X_2,\dots, X_n$ be iid rvs. If there exist constants $a_n > 0$ and $b_n\in \mathbb R,$ and some distribution $G,$ such that $\frac{M_n - b_n}{a_n}\to G.$ There are three extreme value distributions that can be represented using the generalized extreme value distribution family.

  1. Fréchet: $$\Phi_\alpha(x)=\begin{cases}0,&& x\leq 0\\e^{-x^{-\alpha}},&& x>0\end{cases}$$
  2. Weilbull $$\Psi_\alpha(x)=\begin{cases}e^{-x^\alpha}&& x\leq 0\\1,&& x>0\end{cases}$$
  3. Gumbel $$\Lambda(x)=e^{-e^{-x}}$$

Now, the tail of a Fréchet decays as a power law (proven by applying a substitution for $x^{-\alpha}$ and L'Hopital rule):

$$1- \Phi_\alpha(x) = 1 -e^{-x^{-\alpha}}\sim x^{-\alpha}, \; x \to \infty$$

Hence $$\frac{(2x)^{-\alpha}}{x^{-\alpha}}=\small\text{constant}$$ which is the idea behind a fractal structure.

If the tail values of a random variable are in the DOA of a Fréchet, they exhibit power-law mathematical properties, which is equivalent to saying that they have fractal properties.

Generalized Pareto Distribution:

The generalized Pareto GPD, used in the peak-over-threshold approach (Pikands-Balkema-de Haan theorem), considers values in the distribution above a certain threshold.

$$G(x;\mu.\sigma;\xi)=\begin{cases}1-\left( 1+\xi \frac{x-\mu}\sigma\right)^{-1/\xi},&\xi\neq 0\\ 1-e^{-\frac{x-\mu}{\sigma}},& \xi=0\end{cases}$$

with $\xi=1/\alpha$ which would be a type II or III Pareto distribution.

The Pickands-Balkema-De Haan theorem states that if a distribution belongs to one of the maximum domain of attraction of the three extreme value distributions, i.e. Fréchet, Weilbul or Gumble, there is a threshold $u,$ such that if it is large enough the excess distribution can be well approximated by a generalized Pareto.

Fat tails:

A distribution is fat-tailed if it falls within the DOA of the Fréchet. However, there are other criteria for fat tails quoted, such as a distribution without a finite variance. These would include all the stable distributions, except for the Gaussian. For example, the Cauchy distribution is fat tailed, and even its first moment is not defined.

If the Cauchy with location $x_0=0$ and scale parameter $\gamma =0$ has a pdf of the form $\frac{1}{\pi}\frac{1}{1+x^2},$ distributions even more leptokurtic can be derived with expressions of the form $\frac{c}{1+\vert x \vert^k}$ with $1<k<2.$

Likewise, the Pareto with alpha shape parameter of $1$ or $2$ has no finite second moment, since it is a property of the Pareto to have only up to $k$-th finite moments with $\alpha > k$).

The Pareto distribution is the fat-tailed distribution. But not the only one: The Cauchy distribution has no defined moments. Likewise, the Student $t$ distribution spans all tail types depending on the degrees of freedom: a $\nu =1$ $t$ distribution is a Cauchy, while as the $\nu \to \infty$ the distribution becomes a Gaussian.

As an illustration I use the Zipf plot, which serves to analyze the presence of fat tails (linear plot since a power law distribution $\small \bar F_X(x) = \left(\frac{x}{x_0}\right)^{-\alpha} \implies \log\left( \bar F_X(x) \right) = \alpha \log x_0 - \alpha \log x$) to take advantage of finite sample bias at high values of the support in $\small 1,000$ samplings from Gaussian, Cauchy and Pareto and show the instability in the latter two:

enter image description here

Fat tailed distributions (i.e. Pareto distribution) has a survival function $\bar F_X(x)$ that is a regularly varying function (see explanation here), i.e. it can be expressed as the product of a power-law decay and a slowly-varying function:

$$\bar F_X(x) = x^{-\alpha} L(x)$$

A regularly varying function is a function such that $\lim_{x\to \infty}\frac{f(tx)}{f(x)}=t^\alpha$ and if $\alpha=0,$ it is called slowly varying. Since $t^0=1,$ this coincides with the definition in Wikipedia of slowly varying functions: a slowly varying function in which the relative differences in the tail is equal to zero: $\lim_{x\to \infty} \frac{f(tx) - f(x)}{f(x)}=0.$

The proof that the survival function of the Fréchet is regularly varying:

$$ \lim_{x\rightarrow \infty} \frac{1- e^{-(tx)^{-\alpha}}}{1-e^{-x^{-\alpha}}} = \lim_{y\rightarrow 0} \frac{1- e^{-t^{-\alpha}y}}{1-e^{-y}} = \lim_{y\rightarrow 0} \frac{t^{-\alpha}e^{-t^{-\alpha}y}}{e^{-y}} = t^{-\alpha} $$

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  • $\begingroup$ What's wrong with your answer? $\endgroup$
    – Bob Jansen
    Jul 22 at 20:56
  • $\begingroup$ @BobJansen I think it need a lot of work. Give me any suggestions or if you see any errors, please... $\endgroup$ Jul 22 at 22:18
  • $\begingroup$ It's comprehensive (both detailed and broad), relevant links and citations and it addresses the question at hand. +1 from me $\endgroup$ Jul 23 at 7:31
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    $\begingroup$ Math tourist: I believe you did a great job. Definitely worth keeping! $\endgroup$
    – Bob Jansen
    Jul 23 at 18:36

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