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For a distribution $X$, assuming it's "normalized" so that $E\{X\}=0$ and $Var\{X\}=1$.

Given significance level $\alpha$, define "Fat Tail Ratio" as $$FTR(X,\alpha) = z_{1-\alpha/2} - z_{\alpha/2} $$

where $z_{\alpha/2}$ and $z_{1-\alpha/2}$ means $P\{x<z_{\alpha/2}\} = \alpha/2$, and $P\{x<z_{1-\alpha/2}\}=1-\alpha/2$.

Which distributions have the high FTR?

For example, assuming $\alpha=5\%$...

For Normal Distribution, $FTR(N,5\%) = 2*1.96 = 3.92$.

For Student t-distribution, according to wiki page, let $X = \sigma T$, for $v>2$, $Var\{X\} = \sigma^2 \frac{\nu}{\nu-2}$.

So if $\nu=4$, $Var\{X\} = 1$ => $\sigma=\frac{1}{\sqrt{2}}$. Hence $FTR(t(\nu=4), 5\%) = 2 * 2.776 * \sigma = 3.926 $.

I'm wondering if there are other distributions or distribution families, commonly used, has even higher "Fat Tail Ratio"?

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    $\begingroup$ Why is a difference being called a ratio? $\endgroup$
    – Glen_b
    Aug 20, 2013 at 2:39
  • $\begingroup$ let's just say because the distribution is assumed to be normalized. $\endgroup$
    – athos
    Aug 20, 2013 at 4:17
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    $\begingroup$ That in no way deals with the question, it just shifts it one adjective. How is a normalized difference a ratio? $\endgroup$
    – Glen_b
    Aug 20, 2013 at 4:28
  • $\begingroup$ well, i suppose FTR := (z0.95- z0.05) / stdev. so this is a ratio? $\endgroup$
    – athos
    Aug 20, 2013 at 4:40

1 Answer 1

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The Cauchy is a classic one for support on the real line. If you want support on the positive reals, the Pareto distribution is a good one.

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    $\begingroup$ The Cauchy is (i) subsumed by the OP's mention of the $t$ (being $t_1$), and (ii) doesn't have a variance by which to normalize. So either it's outside the OP's conditions, or already covered in the OP's question. $\endgroup$
    – Glen_b
    Aug 20, 2013 at 4:29
  • $\begingroup$ yeah dear Cauchy distribution has infinite even moments... so there's no variance nor kurtosis defined... $\endgroup$
    – athos
    Aug 20, 2013 at 5:51
  • $\begingroup$ Pareto is good but only for positive real number. based on en.wikipedia.org/wiki/Pareto_distribution , $pdf = \beta x_m^\beta / x^{\beta+1}$ , so let $\alpha = 95\%$, $x_m = 1$, $z0.95 = (\beta / 0.95) ^ {1/(\beta+1)}$ => this is approximate to 1: choose $\beta=5$, it is 1.319. Variance is $x_m\beta/(\beta-1)^2/(\beta-2) = 5/48$, so stdev is 0.3227, FTR = 1.319 / 0.3227 = 4.08. Good! "fatter" than normal distribution... $\endgroup$
    – athos
    Aug 20, 2013 at 6:07

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