# How is the tail of a distribution defined (about heavy-tailed distributions)?

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.

• A good working definition is the one you mention; that is, a heavy-tailed distribution is one that decays at a speed that is less than exponential. This agrees with the formal definition from Wikipedia as well: en.wikipedia.org/wiki/Heavy-tailed_distribution which says a distribution is "right" heavy-tailed if $\lim_{t \to \infty} e^{\lambda t} P(X > t) = \infty$ for all $\lambda > 0$. Commented Oct 2, 2016 at 16:59
• The answers here discuss a number of possible ways of defining heaviness of tail, and whuber's answer explains why (in relation to the right tail) one definition makes particular sense. Commented Oct 2, 2016 at 17:03
• MRE, "mean residual lifetime" $\equiv \ E( X - x | X \geq x )$ is an intuitive thing to look at: what's my remaining life expectancy if I'm $x$ years old ? Bryson, "Heavy-tailed distributions: properties and tests" (1974) says "increasing MRE provides a reasonable way of describing the heavy-tail phenomenon" and considers MRE constant, $X$ exponential, vs. $a + bx$. (He calls MRE "CME", conditional mean exceedance.) Commented May 30, 2023 at 16:55

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.