distributionsexponential distributionfat-tailsheavy-tailedterminology
Some distributions are said to be heavy-tailed. It seems that one definition of a heavy-tailed distribution is that its tails are heavier than the tails of an exponential distribution. However, how does one exactly define the tails, since the different distributions have different numbers of parameters?
I suspect that the cumulative distribution function is somehow used here, though I am not sure.
I would say that the usual definition in applied probability theory is that a right heavy tailed distribution is one with infinite moment generating function on $(0, \infty)$, that is, $X$ has right heavy tail if $$E(e^{tX}) = \infty, \quad t > 0.$$ This is in agreement with Wikipedia, which does mention other used definitions such as the one you have (some moment is infinite). There are also important subclasses such as the long-tailed distributions and the subexponential distributions. The standard example of a heavy-tailed distribution, according to the definition above, with all moments finite is the log-normal distribution.
It may very well be that some authors use fat tailed and heavy tailed interchangeably, and others distinguish between fat tailed and heavy tailed. I would say that fat tailed can be used more vaguely to indicate fatter than normal tails and is sometimes used in the sense of leptokurtic (positive kurtosis) as you indicate. One example of such a distribution, which is not heavy tailed according to the definition above, is the logistic distribution. However, this is not in agreement with e.g. Wikipedia, which is much more restrictive and requires that the (right) tail has a power law decay. The Wikipedia article also suggests that fat tail and heavy tail are equivalent concepts, even though power law decay is much stronger than the definition of heavy tails given above.
To avoid confusions, I would recommend to use the definition of a (right) heavy tail above and forget about fat tails whatever that is. The primary reason behind the definition above is that in the analysis of rare events there is a qualitative difference between distributions with finite moment generating function on a positive interval and those with infinite moment generating function on $(0, \infty)$.
I applied Tukey-Lambda PPCC to your data to get the following plot:
You see the local max at $\lambda=-0.47$, which implies heavier tails than normal distribution. For normal it's 0.14 and for Cauchy it's -1. So, if you go with this view, then your data's tail is not as "fat" as Cauchy's, but much fatter than normal.
However the global max is at $\lambda=33.17$
I drew the Tukey lambda for the above two maxima here:
So, you have two very different views of the distribution. One is fat tailed, the other one's very convex.
I think that the convex distribution is the better fit. Cauchy and other fat tailed ones are not a good fit here, go for these weird distributions like in my last plot.
Here's how Tukey Lambda distribution looks like for some $\lambda$.
I use this tool to gauge the shape of my data.
Best Answer
We distinguish what distributions are heavy tailed by first limiting our discussion to those tails that are long, that is, there is always an $\epsilon>0$, no matter how small, for which $f(x)>\epsilon>0$ for any $x<M$ no matter how large $M$ (for right tails), or $x>M$ for $M$ large magnitude negative (for left tails). In other words, $f(x)$ is non-zero no matter how large $|x|$ is. A random variable rather than density function definition for long tailed would be equivalent.
Then (using the right tail) $\lim_{x\rightarrow \infty} 1-F(x)\rightarrow 0$, that is, the long heavy-tailed survival function, i.e., $1-F(x)$, A.K.A. $1-\text{CDF}$, can then be used to construct the ratio of two candidate survival functions, which ratio will go to zero as $x\rightarrow \infty$ if the lighter tail is in the numerator. In practice, it is often easier to compare the limiting logarithm of the ratio of survival functions, but this is actually not different, if properly interpreted. For long left tails, we would compare the limiting (logarithm of) the ratio of the CDF's themselves as $x\rightarrow -\infty$, rather than the survival functions.
Why use the CDF or 1-CDF for this? Why not use the (e.g., logarithm) of the ratio of the pdf's? In many cases we could use the pdf's, however, for actual random variables (observations), and for some pdf's with nasty properties like non-smooth derivatives, this would be less revealing than comparison of the limiting areas under the pdf's.
What is the big deal with tail heaviness of exponential functions? Exponential functions have the same rate everywhere so that they are memoryless. Thus, exponential distributions form a natural cut point for measuring heaviness of tails.