I think the title is self-explanatory. I understand that the skewness and the tail behavior of some distribution are completely unrelated as any symmetric distribution will have a skewness of zero irrespective of how heavy its tails are.

However, I was wondering (i) if the skew-normal distribution and the skew-Cauchy distribution are heavy-tailed.

(ii) Also, we can get skewed distribution using a selection procedure. That is, if $X$ and $Y$ are Gaussian (or Cauchy), $\frac{d}{dz}P(X-kY\leq z|Y>0)$ represents a density of a skewed version of normal (or Cauchy distribution). What about the tails of such constructed skewed distributions?

Note: I say the distribution of a random variable $X$ heavy-tailed if $\lim \limits_{x\rightarrow \infty} e^{tx} P(X>x) = \infty$ for all $t>0$.

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    $\begingroup$ Which definition of heavy-tailed are you using? $\endgroup$ – Glen_b Jun 30 '19 at 12:16
  • $\begingroup$ Hi @Glen_b, I only know one definition of heavy-tailed distributions. That is, the distribution of a random variable $X$ is said to be heavy-tailed if $\lim \limits_{x\rightarrow \infty} e^{tx} P(X>x) = \infty$ for all $t>0$. $\endgroup$ – Joy Jun 30 '19 at 12:19
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    $\begingroup$ That's fine, though some people mean other things when they say "heavy tailed". Could you edit that into your question, please? $\endgroup$ – Glen_b Jun 30 '19 at 12:20
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    $\begingroup$ Are you using this skew Cauchy and this skew normal, or something else? $\endgroup$ – Glen_b Jun 30 '19 at 12:24
  • $\begingroup$ Yes, I am using the mentioned skew normal and skew Cauchy. $\endgroup$ – Joy Jun 30 '19 at 12:27

I'll take it that you're happy to take it as a given that the ordinary (symmetric) normal is light tailed and the ordinary (symmetric) Cauchy is heavy tailed.

That the indicated skew-normal is not heavy-tailed is easy to see. I'll discuss the 'standard case' of the skew-normal; the scale-location case is not really any more work.

Taking $f(x)=2\phi (x)\Phi (\alpha x)$, we can immediately see that $f(x)<2\phi(x)$, so $P(X>x)<2(1-\Phi(x))$. Indeed, we can see that $\lim \limits_{x\rightarrow \infty} e^{tx} P(X>x)$ is less than twice that of a standard normal (which we already know to be light tailed, so we know that integral is finite even before we try to evaluate it).

I was somewhat surprised that you selected the linked skew-Cauchy (there are a number of others), because it's associated with a different skew-normal (which can be found within the article I linked to).

A somewhat similar approach to the one I used for the skew-normal can be used. Let's restrict consideration to the right half (i.e. consider only $x>\mu$). Then the density is proportional to the density of a Cauchy (because of the parameter $\lambda$), so $P(X>x)$ is proportional to that for a Cauchy. Consequently, the limit goes to infinity because the corresponding quantity for the Cauchy does.

So (without actually having to evaluate it in either case) that particular skew-normal is light tailed and that particular skew-Cauchy is heavy tailed.


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