Re: Gamma and other right-skewed distributions #2Dear Madam
I have the StatsDirect statistical package. It allows me to generate random numbers in 17 distributions of which the normal and lognormal I know do not fit, the uniform(both versions), Logistic, Student-t(squared?), and Cauchy seem too blilateral to fit, the gamma is what I have already tried and the exponential and chi-squared are special cases of gamma, Poisson, Binomial, Geometric, Negative Binomial and FRatio(variances) are not listed in the Compendium of Common Distributions for me to check, and beta and Weibull both seem possible for a right-skewed distribution.
Again is there any sort of decision tree in existence which would alllow narrowing down possible distributions from scratch? Or is there any software which will calculate parameters a data set would have if it were in each common distribution, generate random numbers for each distribution and its parameters, and provide a measure of how the actual data fit with the random nmbers for each distribution(lsum of squared differences?)
Has anyone done anything on how exhaustive lists of common distributions such as Johnson and Kotz's are? How much possible distributions can deviate from the nearest known distribution? Or how much recorded distributions do?
From the Johnson and Kotz books I see that there are systems of distributions, but none seems to correspond to right-skewed distributions as a class.
I am mainly interested in the distribution of one ratio of parameters and what theoretical information I can derive from that, but would also be interested if there were more parametric( using-more-of-the-data) ways of comparing it to the (log-normal and power-law) distributions of other measures of the same sample than the Kendall's taus I already have.
The Compendium gives the mean of gamma as BC (the parameters found in variance) +A (a location parameter) where Wikipedia gives it as just k times theta.
The fit of the plot to the gamma distribution is so close that I doubt there is much to be found in trying other distributions but I am intersted in the question of determining distributions generally. MathSciNet returns 5 items for "determining distribution*" anywhere. Of the 3 that seem relevant, one says things would be easier if there were a natural law that helped determine distributions, one is on distributions of "time to failure" and one is a review of Janos Galombos's Introductory Probability Theory one of which's unusual features is a section on determining distributions. They give very different impressions of how well mastered this task is by the science of statistics.
Thank you for your activity and help to my mathematically illl-informed gropings
Yours Sincerely,
Alan E. Dunne
----- Original Message -----
From: kornbrot
To: Alan Statistics
Sent: Wednesday, June 13, 2007 2:57 AM
Subject: Re: Gamma and other right-skewed distributions #2
Do you have access to a statistical package?
My recommendation would be to use SPSS ro similar to fit to selection of skewed distirbutions
It is useful if when you pose question you briefly describe your actual problem.
For example, it matters if you are just looking at 1 group and fitting gamma or whatever, or if you wish to compare groups
Wikipaedia is not best source in this case, since referneces are slightly out of date and ONLY to gamma
The acknowledged classic is
Johnson, N. L., & Kotz, S. (1970). Continuous distributions in statistics (vol. 2). Boston: Houghton Mifflin Co.
.
There is later edition
diana
On 11/6/07 19:14, "Alan Statistics" <[log in to unmask]> wrote:
Dear Madam
I used the ratio of variance to mean, and the ratio of the mean to that to estimate alpha and beta, working from the values given in the Wikipedia article on the gamma distribution.
I have no theoretical grounds for expecting the distribution to be a gamma one. I'm affraid I simply heard of gamma as a common right-skewed distribution different from the lognormal and power-law ones I knew didn't fit, and tried to find a way of checking if that particular distribution fit -particularly when I read that it could underlie L-shaped distributions. I just compared the random gamma version by eye to the actual one but the fit was so close I don't think it could be minimised further. Is there a way of working through possible distributions by some sort of decision tree?
I have looked at the article by Dr Hosking. I was unable to understand much, particularly not what "L-moments" actually are, mainly because of my lack of mathematical grounding. I did grasp that the method for determining distributions still involves picking likely distributions and generating random measures from them to compare to the actual ones His example, further, involves estimation of distributions from multiple samples, while I have only one.
Yours Sincerely,
Alan E. Dunne
----- Original Message -----
From: kornbrot <mailto:[log in to unmask]>
To: Alan Statistics <mailto:[log in to unmask]>
Sent: Saturday, May 12, 2007 3:14 AM
Subject: Re: Gamma and other right-skewed distributions #2
What do you want to know?
Is it that you want to know the best theoretical distribution for your data?
Or do you have theoretical reason to believe it is gamma, and want to find best fitting parameters?
How did you estimate gamma parameters? Did your use skew and kurtosis or some best fit methodthat, say minimised the K-S parameter? Or the departure from linearity of the Q_Q plot?
If you are using skew and kurtosis, I would recommend linear versions, not well known b ut the bets I have found
Hosking, J. R. M., & Wallis, J. R. (1997). Regional frequency analysis. An approach based on L-Moments. Cambridge, UK: Cambridge University Press.
Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, B, 52, 105-124.
It is very good on how to get the parameters of a whole lot of dsitributions
Best
Diana
On 12/5/07 03:29, "Alan Statistics" <[log in to unmask]> wrote:
With Respect
The most helpful general references I got were to Wikipedia(starting from http://en.wikipedia.org/wiki/Gamma-distribution) and to N.L. Johnson and Samuel Kotz Continuous Univariate Distributions.
I thought my ratio of rwo variables approximated to the shape of one exponential distribution, a special case of gamma.But trying to estimate parameters has shown me that if it is a gamma distribution I have it is one with k<1; the shape is more like the exponential than any of the higher-k gammas, but differs from it in the way opposite to them.
Three-parameter gamma is referred to in message 017726 on this list(05/12/05)
I have used StatsDirect to create a random-numbers gamma distribution with the parameters I calculated from my data.I have compared the two both with a Smirnov test, whose results I was unsure how to interpret, and by plotting points with one as x-axis and the other as y-axis, which gave me almost a straight diagonal line (with small kinks).
Yours Sincerely,
Alan E. Dunne
> I will answer part of your question.
> For me a typical Gamma dist is one whose density initially increases and then
> decreases. I am not sure if when you talk about three parameter gamma,
> you mean
> the variant with a shift (i.e. instead of (0,infinity) look at (a, infinity)
> and "a" is the third parameter in this case but it obviously does not change
> the shape of the didtribution.
>
> To test if the distribution of your data is a gamma (or any other
> distribution)
> , use probability plots (there ar evariants called QQ plots or similar).
> Most statistical software provide these. Menu driven software also
> will let you
> specify the distribution (gamma in your case) and may be the parameters.
> Roughly speaking, if the plot "resembles" a straight line, than you are
> spot on.
> These plots are usually accompanied by p-values or similar statistics so that
> you may use also a traditional statistical judgement.
> Most of the statistics should be interpreted cautiously if the
> parameters of the
> distribution are estimated from the data (rather than supplied by you) but
> otherwise I have found them very useful for the task.
>
> You don't mention particular books but probably the volumes of Johnson
> and Kotz
> are the best place for initial reference.
>
> Best regards,
> Georgi Boshnakov
>
> ============================
> Georgi Boshnakov
> School of Mathematics
> University of Manchester
> Manchester M60 1QD
> UK
Alan,
I saw your allstat posting. I find www.xycoon.com a useful resource for density function and other formulae relating to statistical distributions with which I am not familiar - it has a good section on continuous distributions and includes relationships among them. This may be obvious, but watch out that you don't confuse the "gamma distribution" for the "gamma function" (gamma(x) = (x-1)! where x is an integer).
Dominic Muston
Hi
For general info on distributions try wikipedia
e.g. http://en.wikipedia.org/wiki/Gamma-distribution
>> How can I test whether my dataset approximates a gamma distribution?
To test whether two datasets come from the same distribution use a Kolmogorov-Smirnov test.
If one of the datasets is your real test data, and the other is sampled from the gamma distribution, then you have your test.
K-S tests come in two forms (in stata)
2 sample - your supply two datasets
1 sample - you supply your test dataset, and specify the distribution they are to be tested against.
Desmond
Respect for digging up so much literature to understand Gamma
distribution. There's a lot in your email that I don't know. But in my
diploma in Statistics, we did learn something simple which you didn't
seem to have mentioned. The Gamma distribution represents the
distribution of time until the n events if events occur with constant
rate. The Exponential distribution is thus a special case of the Gamma
as it's the distribution of time until the occurance of the first event.
The Chi-square distribution is also a special case of the Gamma.
Regards,
Tim
[Timothy Mak]
Professor Diana Kornbrot
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Professor Diana Kornbrot
Evaluation Co-ordinator, Blended Learning Unit
University of Hertfordshire
College Lane, Hatfield, Hertfordshire AL10 9AB, UK
email: [log in to unmask]
web: http://web.mac.com/kornbrot/iweb/KornbrotHome.html
Blended Learning Unit
voice +44 (0) 170 728 1315
fax +44 (0) 170 728 1320
Psychology
voice +44 (0) 170 728 4626
fax +44 (0) 170 728 5073
Home
19 Elmhurst Avenue
London N2 0LT, UK
voice +44 (0) 208 883 3657
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