[ArXiv] Pareto Distribution
Astronomy is ruled by Gaussian distribution with a Poisson distribution duchy. From time to time, ranks are awarded to other distributions without their own territories to be governed independently. Among these distributions, Pareto deserves a high rank. There is a preprint of this week on the Pareto distribution:
On the Truncated Pareto Distribution with applications by Zaninetti and Ferraro [astro-ph:0804.0308]
From the abstract:
This note deals with an application of the Pareto distribution to astrophysics and more precisely to the statistical analysis of mass of stars and of diameters of asteroids. In particular a comparison between the usual Pareto distribution and its truncated version is presented.
The paper introduces the pdf, cdf, mean, variance, higher moments, and survival function of the (truncated) Pareto distribution with applications to Star masses from the Hipparcos data[1] and asteroid sizes, and simulations of primeval nebula[2]. It concludes that the truncated Pareto works better than the usual Pareto. The Pareto distribution is simple and intuitive.
ps. Not many astronomy papers cite papers from recent statistical publications. I witness that although the most of astronomical papers have no needs for citing papers in statistics, if they do, they tend to have references from four to five decades ago among which books were revised in 90′s or later and articles of modern perspectives are available (exceptions are seminal papers that introduced statistics to the community like EM algorithm). It is quite encouraging to see an article from JASA 2006 was cited in [astro-ph:0804.0308]
hlee:
A review of arxiv:physics.data-an:0706.1062 on power laws from the slog:
04-07-2008, 6:51 pmEverything you wanted to know about power-laws but were afraid to ask
TomLoredo:
Nice post, Hyunsook. The Pareto distribution (and its relatives) appears all over in astronomy, but almost never with the name so familiar to statisticians. Instead, you see it as: power-law distribution, truncated power-law, Salpeter initial mass function, Schecter function (really a kind of gamma dist’n, but in a regime where it looks like a truncated power law). Roughly speaking, a power law is an indication of some kind of scale-free behavior. The slope can be an important indicator of the underlying physics, and astronomers often don’t do a great job in estimating it (esp. when there is measurement error). But truncation or “breaks” (change in the Pareto power law index) are often the most exciting thing, as they indicate the presence of important physical scales in the underlying process. So astronomers do often find themselves trying to infer breaks and truncation. For truncation in particular, the statistics is messy (I had to look into it for inferring the edge of the Kuiper belt in a paper some years ago). The likelihood is “non-regular” (i.e., the usual asymptotics doesn’t hold), and there is still research being done in statistics on how to do things like calculate sound confidence or credible regions.
04-11-2008, 3:56 pmhlee:
A few questions I wish to address. My vague memory tells that this Pareto distribution appears in economics to explain the 80% wealth is distributed among 20% people. The broken power law implies this wealth distribution is different before and after the breaking point, whereas the truncated power law includes an upper limit into fitting process but the wealth allocation is still same. My questions are how to test whether the distribution is a power law, a broken power law, or a truncated power law, how to test one of these power laws is a good fit, and how to determine the breaking point is significant regardless of the righteousness of the broken power law. The former question also asked for model selection or variable selection issues, in which so far physics has played a dominant role to choose among three may be more for multiple broken points but I wonder if statistics can contribute further.
I really do appreciate your telling different nomenclatures so that I can look for those names with care in future.
04-14-2008, 8:33 pmvlk:
A little digression away from Astronomy.. even in Economics the Pareto can be a diagnostic tool. If there is no scale in the system, then that system will show a self-similar or power-law distribution. This happens, for example, in feudal societies where there is no governmental authority to redistribute wealth via targeted taxation. (I recall seeing a paper recently which plotted the wealth and income in medieval Hungary, and that was a power-law far as the eye could see. Ask if you want to see that paper and I will try and hunt it down.) In modern societies, it seems the lower income groups are no longer power-law distributed, though the upper income groups still are. (If you google for Pareto or power-law and income distributions you will find a whole lot of work that has been done on this.)
04-19-2008, 3:58 pm