# What is this distribution? (time-delays between two events)

I have a right-skewed distribution that has this qq-plot against a normal with the same mean and standard deviation:

The data are time-delays between two events. Also, this is the histogram of the data:

and this is the QQ-plot of log(data + 1), vs. normal of the same mean and standard deviation as the log(data+1) values:

I have tried fitting Weibull, lognormal, and Pareto, without success. Any thoughts on what would be the right candidate?

• How did you try fitting these distributions? You might also post a histogram, some more details about what kind of data this is (that might affect the distributions/methods proposed), and what the purpose of your analysis is. The more information the more people here can do to help. With that said, if you don't know that is is a standard distribution then it would be a breeze to fit a mixture of Normals to it. – user44764 Jun 9 '14 at 4:18
• It's very difficult to assess which distribution something is, but easy to tell what it isn't (it isn't normal!). At best you might hope to identify some distribution that might give a reasonable description. If the values are all $>0$, what does a QQ plot of the logs look like? – Glen_b Jun 9 '14 at 4:38
• @Matthew, you lost me. I can't see how one could fit a finite mixture of normals to something like this. – user765195 Jun 9 '14 at 4:53
• The rounding can have odd effects, which you may need to worry about. – Glen_b Jun 9 '14 at 5:29
• I just took a guess at the shape parameter - it seemed from your plot it should be less than 1, and I tried 0.1, 0.2 and 0.5, any of which looked roughly right, but the shape wasn't quite there - I couldn't get both ends right at the same time. I tried a couple of mixtures and then added a shift to get something vaguely like the right shape. Essentially trial and error (with a bit of knowledge at the start) to get a rough approximation. No doubt you could do better. – Glen_b Jun 9 '14 at 6:00

Taking your second Q-Q plot, it gives us some clues - if it were lognormal, the upper right part would look straight.

So that suggests that perhaps it's something lighter-tailed than lognormal, at least for the larger values.

A right-skew distribution that's lighter-tailed than lognormal is the gamma. It seems pretty skew - substantially more skew than an exponential for example, so it suggests a shape parameter smaller than 1.

Here's an example:

x=c(rgamma(1000,.1,1/9000))
x1=round(x);qqnorm(log(1+x1))


However, it's difficult to get the shape of the Q-Q plot quite right; one possibility is that a mixture of two gammas might do better.

Another thing to consider is shifted gamma distributions - adding something like 0.4 before rounding does seem to improve the shape at the low end.

Experiments with mixtures does suggest that a finite mixture with a few components might do quite well, so it's probably worth trying a 2 or 3 component mixture of gammas or shifted gammas (or indeed similar less-skew-than-lognormal distributions) and using a decent fitting algorithm, rather than the bit of trial and error I used to get this far.