Regarding the lognormal, in many applications one can justify it if the central limit theorem is applicable to the logarithm of the variable of interest--see, e.g., http://www.mlahanas.de/Math/Lognormal.htm. I've also seen a lognormal error model derived from a stochastic differential equation for fatigue cracking in metal.

You could also think about the Weibull distribution; this is a distribution of minima, so it applies to situations where you have a "weakest link" model, e.g., the breaking strength of chains.

Here is a paper that derives the generalized gamma distribution (of which the gamma and Weibull are both special cases) from a physical model: http://www.uh.edu/engines/generalizedgamma.pdf. Presumably you could also derive your vanilla gamma from a simplification of this.

Derivations of the gamma from a physical process seem to be less common than for some other distributions. Maybe if you post some specifics about your application someone will have a better answer.

Regarding the lognormal, in many applications one can justify it if the central limit theorem is applicable to the logarithm of the variable of interest--see, e.g., http://www.mlahanas.de/Math/Lognormal.htm. I've also seen a lognormal error model derived from a stochastic differential equation for fatigue cracking in metal.

You could also think about the Weibull distribution; this is a distribution of minima, so it applies to situations where you have a "weakest link" model, e.g., the breaking strength of chains.

Here is a paper that derives the generalized gamma distribution (of which the gamma and Weibull are both special cases) from a physical model: http://www.uh.edu/engines/generalizedgamma.pdf. Presumably you could also derive your vanilla gamma from a simplification of this.

Derivations of the gamma from a physical process seem to be less common than for some other distributions. Maybe if you post some specifics about your application someone will have a better answer.

Hi, I do not have a definite answer. Try this kind of reasoning and see if it applies to your situation: given mean and std deviation, the max entropy solution is a gaussian. Now, the gamma distribution is the maximum entropy probability distribution for a random variable X for which E(X) is fixed and greater than zero, and E(ln(X)) is fixed ( see wikipedia for details, the symbols do not show in this window). Do you have good information on the mean of the log of your random variable in your problem?

I explain the success of Gamma distribution in applications as follows. It is the simplest distribution which has power tail for small values, i.e., self-similarity property, and exponential tail for large values, i.e., memory-less property. More precisely one can say it in terms of limit conditional distributions.

Hello Dr. Wilhelm,

Your question gives me the impression that you actually seek for a physical basis for the use of the gamma distribution in modelling right-skewed random in variables in the positive real line.

I'm not a statistician [yet], I'm a biologist. I feel like I should share your concern about the scientific interpretation of the models we use. But frankly I don't. In my undergraduate's and insignificant opinion, statistical distributions are mathematical devices. All this means is that for some of them (Poisson, Gaussian, Exponential) we HAPPEN to have a somewhat physically interpretable explanation. As you mentioned, the gamma distribution arises when one considers a particular problem regarding the waiting times of a Poisson R.V. In some particular applications, for instance, when we want to know how many minutes we have to wait until the next call is made given that we had 5 calls the past hour, the gamma distribution has a clear physical meaning. Many other times, it is just convenient because of its mathematical flexibility.

In short, what I want to contribute to your thought-provoking question is that, for me at least, the theoretical reason why to use a gamma distribution in some modeling problem has nothing to do with physical interpretation but indeed to its mathematical properties.

Despite my personal opinion, I think you concern about physically-based derivation is rather valid. The article below seems to bear in the same phylosophical orientation and, more interestingly, concerns a related distribution, the generalized gamma.

I hope my comment adds a complementary perspective to the topic.

Best,

http://www.uh.edu/engines/generalizedgamma.pdf

I am not certain I am really answering your question but as a clinical biochemist, who studies biomarkers, I come across right skewed data all the time. The underlying mechanism for explaining the distribution is that the body normally holds most (blood/cellular) proteins within a defined range for health young interviduals. As we age the mechanisms for producing and holding the biomarkers within the set limits are less able and strict such that we see a gradual shift in higher levels. The major right skewing occurs when we have disease and the deregulation,( which is nearly always positive - too much of something as we cant measure negatives in real life) results in huge variable positive numbers - it is like random numbers a let free in short bursts .

However - as an aside comment from a simple biochemist I do not like fitting distributions to such data and always use non-parametrics in my analysis. Not particularly good for modelling processes but....... I think such fitting does not accurately describe the data and underlying process in biology.

I think you check the suggestion of Collins i.e paper " A physical basis for Generalized gamma " that is available in http://www.uh.edu/engines/generalizedgamma.pdf

If distribution is skewed use higher order cumulants to estimate parameters of time series

You seem to have per-conceived ideas about error distributions, and my own opinion is that the normal distribution is an over-achiever, as Mr. Dr. Prof. Bradley Efron proclaims. Not being a normal person, it's hard to defend normality, but is generally the least of our worries.

Call for Papers:

"12th International Probabilistic Workshop" held in Weimar Germany, on November 4th and 5th 2014"

https://www.researchgate.net/post/Call_for_Papers_12th_International_Probabilistic_Workshop_held_in_Weimar_Germany_on_November_4th_and_5th_2014

error distributions can also be mapped using link models like logit, probit, cauchy etc. The continuous log log model is associated with a right skewed distribution while a normal logit or the logistic function can be used as a starting ponit for a model with non normal errors in Extreme value distributions., providing a logical choice. However these are discrete choice models with wide applications.