Talk:Inverse Gaussian distribution
|WikiProject Statistics||(Rated Start-class, Low-importance)|
it would be illustrative to have a graphical represaentation of the distribution
The last change (November 4) to the Brownian motion section was made by me... reference is "The Inverse Gaussian Distribution: Theory, methodology and applications" p29.
Deavik 23:39, 4 November 2007 (UTC)
There are several parameterizations of the inverse Gaussian distribution, including one which makes the relationship between it and a Brownian motion with drift more explicit (in terms of the drift parameter v and the variance parameter ). These other parameterizations should be at least mentioned. I am not aware of any good reason to present the given one as "canonical"; though if such a reason exists, it, too, should be presented. Cheers, Eliezg 05:04, 6 November 2007 (UTC)
Error in likelihood function.
I think the likelihood function should be changed to:
Without this additional term, you can not solve the first order condition for the MLE's, as they are given in this article.
the misleading part
the article mentioned the name is misleading, and the inverse gaussian distribution is not the distribution of (X is normal), then what is the name of the distribution of Y???? It will be great if we say something about that. Jackzhp (talk) 14:11, 28 July 2009 (UTC)
- If X is normal, the distribution of 1/X doesn't seem to have a special name, but its pdf is straightforward to work out from probability density function#Dependent variables and change of variables. I can't find any references but from a quick bit of experimental mathematics i think it doesn't have any moments if the mean of X is zero and none apart from the mean otherwise, and it's oddly-behaved around zero in general, although if the coefficient of variation of X is large you may not notice in practice. See also Box-Cox distribution for the case when X is truncated at zero. As i don't have any references, it's rather messy and doesn't seem to have much theoretical or practical importance, i'm not going to add anything to the article myself. Qwfp (talk) 18:01, 25 May 2010 (UTC)
Error in the labels of the figure
Apart from formatting differences, a figure identical to the one shown can be generated in R using
plot(x, dinvgauss(x, 1, 1), type="l", xaxs="i", yaxs="i", xlab="", ylab="", col=1); lines(x,dinvgauss(x,1,0.2),col=2); lines(x,dinvgauss(x,1,3),col=3); lines(x,dinvgauss(x,3,1),col=4); lines(x,dinvgauss(x,3,0.2),col=5)
However, according to the R manual (e.g., ), the second parameter of the function dinvgauss corresponds to 1/lambda, instead of lambda. Hence, the figure's labels should be replaced. Alternatively, use the following code to generate an interesting sequence of densities:
x<-seq(0,3,0.01); plot(x, dinvgauss(x, 1, 1/4), type="l", xaxs="i", yaxs="i", xlab="", ylab="", col=1,ylim=c(0,2.5)); lines(x,dinvgauss(x, 1, 1/2), type="l", col=2); lines(x,dinvgauss(x, 1, 1), type="l", col=3); lines(x,dinvgauss(x, 1, 2), type="l", col=4); lines(x,dinvgauss(x, 1, 4), type="l", col=5); lines(x,dinvgauss(x, 1, 8), type="l", col=6); lines(x,dinvgauss(x, 1, 16), type="l", col=7); lines(x,dinvgauss(x, 1, 32), type="l", col=8)
Clarifying the initial description
I think one of the introductory paragraphs is likely to be misconstrued, specifically the clause, "while the Gaussian describes the distribution of distance at fixed time in Brownian motion . . . ." It is so commonplace to generalize Brownian motion to higher dimensions that I think it is confusing to claim the process's distance is Gaussian (since in higher dimensions, it isn't). I think this article should not assume that readers be familiar with Brownian motion. If we ditched the word "distance" and replaced it with "level," I think we would not only clear up the ambiguity, but the wording would nicely complement the following sentence: Gaussian describes level at a certain time; inverse Gaussian describes time to hit a certain level. Disadvantage: "level" isn't very precise, although my gut tells me it is fairly intuitive. LandruBek (talk) 22:07, 6 April 2010 (UTC)
If there is no drift...
The article should, I think, also mention the special case of this distribution as μ tends to infinity. This is how I was first introduced to the IG distribution -- it describes the hitting time of a drift-less Wiener process at level λ². The derivation is an elementary one from the CDF of a Gaussian using a simple change of variables, and thus I think readers are likely to expect to see this case mentioned explicitly. LandruBek (talk) 22:07, 6 April 2010 (UTC)
What are the terms occurring in some of the equations? Weights?