Why do there appear to ‘almost’ never be any super-light-tailed distributions? The tail of a normal distribution decays ~exponentially. The only thing that decays super-exponentially offhand that I can think of is aging[1].
I’d suggest at least part of it is that any super-light-tailed distribution more light-tailed than the normal works out in practice to simply looking like it has a hard bound or a range due to rarity and the resulting difficulty of verifying any such rare instance. You would not see them, and you would not know if you did: so for almost all intents and purposes, any super-light-tailed distribution’s “tail” is just a hard cutoff or threshold or range or bound. And we see those all the time.
Even for the normal or Gompertz, there’s a point at which you can safely say “the upper bound is X; you can wait the age of the universe, and you’ll still never see a >X”. Where are the 10-foot tall people, much less the 20-foot tall people? If you didn’t have enormously detailed and large records of human life-expectancies, who would blame you for saying “the Almighty hath ordained the postdiluvian span of man’s year at naught more than six score of years”?
Now, suppose life-expectancies were ultra light-tailed; this would mean in practice something like a ‘Jeanne Calment’-level outlier would not survive into her 120s, instead, she would drop dead 2 minutes after her 80th birthday instead of, like her rivals, dropping dead 1 minute after their 80th birthday. How would you know that, instead of saying, “hm, I guess the watch of the nurse recording her birth certificate was a bit slow” (as you would correctly say in the 99 other such cases which were false positives)?
Perhaps in a physics experiment which has been exactly quantified and where you’ve spent decades systematically eliminating every source of noise, you can go ‘oh my god, a particle decayed after 1.01 picoseconds instead of 1.00 picoseconds! This changes everything—it’s not a hard bound, it’s a super-light-tailed distribution, just as predicted by Katz et al 1971!’ But not anywhere else.
(As an aside: I went ‘no, you seem to be conflating light-tailed with low variance’, wrote up a response, thought about it, and then realized you were absolutely correct.)
This is perhaps a silly question:
Why do there appear to ‘almost’ never be any super-light-tailed distributions? The tail of a normal distribution decays ~exponentially. The only thing that decays super-exponentially offhand that I can think of is aging[1].
Double-exponential in the tail—https://en.wikipedia.org/wiki/Gompertz%E2%80%93Makeham_law_of_mortality.
I’d suggest at least part of it is that any super-light-tailed distribution more light-tailed than the normal works out in practice to simply looking like it has a hard bound or a range due to rarity and the resulting difficulty of verifying any such rare instance. You would not see them, and you would not know if you did: so for almost all intents and purposes, any super-light-tailed distribution’s “tail” is just a hard cutoff or threshold or range or bound. And we see those all the time.
Even for the normal or Gompertz, there’s a point at which you can safely say “the upper bound is X; you can wait the age of the universe, and you’ll still never see a >X”. Where are the 10-foot tall people, much less the 20-foot tall people? If you didn’t have enormously detailed and large records of human life-expectancies, who would blame you for saying “the Almighty hath ordained the postdiluvian span of man’s year at naught more than six score of years”?
Now, suppose life-expectancies were ultra light-tailed; this would mean in practice something like a ‘Jeanne Calment’-level outlier would not survive into her 120s, instead, she would drop dead 2 minutes after her 80th birthday instead of, like her rivals, dropping dead 1 minute after their 80th birthday. How would you know that, instead of saying, “hm, I guess the watch of the nurse recording her birth certificate was a bit slow” (as you would correctly say in the 99 other such cases which were false positives)?
Perhaps in a physics experiment which has been exactly quantified and where you’ve spent decades systematically eliminating every source of noise, you can go ‘oh my god, a particle decayed after 1.01 picoseconds instead of 1.00 picoseconds! This changes everything—it’s not a hard bound, it’s a super-light-tailed distribution, just as predicted by Katz et al 1971!’ But not anywhere else.
Good point.
(As an aside: I went ‘no, you seem to be conflating light-tailed with low variance’, wrote up a response, thought about it, and then realized you were absolutely correct.)