For example the distribution of weights of human. There are not many adults under 40 kg, but a lot more people heavier than 100 kg, although the average of an adult's weight is, let's say, 70 kg. Another example is this human reaction time, sharing the same tail property. I can image a "mixture" of normal distribution and Cauchy distribution, where the normal distribution dominates at the left side and the Cauchy distribution dominates at the right side, but is there already a family of such distributions with good properties, like, that we can easily calculate the mean, variance, entropy, etc?
$\begingroup$
$\endgroup$
5
-
$\begingroup$ Almost by definition, you are asking for distribution families whose elements have nonzero skewness (standardized third moment). I am wondering whether (a) you have a specific, quantitative characterization of "light" vs. "heavy tails" and/or (b) you have additional criteria that would narrow down the (huge) range of possibilities. If not, this question seems to admit so many answers that it may be too broad in scope for this forum. $\endgroup$– whuber ♦Jan 25, 2015 at 21:24
-
$\begingroup$ @whuber Yes, but I am particularly interested in the distributions that 1) have closed forms of properties like lower order moments or entropy; 2) the parameters can be related to the generative process of the data. For example in gamma distribution, we have parameters like shape and scale, but interpreting them as part of the data generation may be less obvious. So I am looking for distributions that people have used for these scenarios, where they at least have good interpretations of the parameters. $\endgroup$– ZiyuanJan 26, 2015 at 13:44
-
$\begingroup$ That is still very broad. Although we could just give you a list of many commonly used distributional forms, these are readily combined into infinitely more in myriad ways, such as with mixtures. This process can still lead to closed formulas and can be described in a "generative" form. You therefore seem to be asking about how to address a large nebulous class of problems. Why not tell us about the specific problem you actually have so that people can suggest effective choices of distributional model for it? $\endgroup$– whuber ♦Jan 26, 2015 at 16:33
-
$\begingroup$ @whuber I don't have a specific problem. I came across the site of human reaction time, and the distribution in the web page reminded me of the human weight distribution which my teacher took as example when she was warning us of the danger of "assuming everything normally distributed", so I wonder whether anything has been done on these cases. $\endgroup$– ZiyuanJan 27, 2015 at 0:48
-
$\begingroup$ Maybe Gumbel Distribution. $\endgroup$– Amarpreet SinghAug 3, 2018 at 18:06
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
There are infinitely many distribution families with heavier right tails and lighter left tails.
A few are well known. I'll mention a few, but don't by any means think this would be all. Some of these have indeed been used for things like reaction-times
Distribution Simple form for
Family Mean Var Entropy
Gamma Yes Yes Yes
Lognormal Yes Yes Yes
Chi Yes Yes Yes (not χ², but its square root)
Fréchet Yes Yes Yes
Weibull Yes Yes Yes (not all members are right skew)
Gumbel Yes Yes Yes
log-logistic Yes Yes ?
F Yes Yes ? (Fisher-Snedecor F)
Skew Normal Yes Yes ?
ExGaussian Yes Yes ?
Here "simple form" means "is available in a form that may be readily calculated with common software", so things like Gamma functions count as 'simple'. In the above table "?" means "I didn't quickly locate one" (which suggests that simple forms may either not exist or be hard to calculate)
See the list of distributions at Wikipedia here -- the individual pages generally list the mean and variance and sometimes give the entropy.
