# Finding analytic form for distributions using linear regression, need ideas

I'm trying to find an analytical form to describe these probability density functions: I'm pretty new to all of this, but think I should use some linear combination of basis functions (so I can then use linear regression to find the coefficients for each of the pdfs) but I'm not sure where to start. It also needs to be relatively easy to integrate analytically when convolved with a gaussian.

$$\int_{-\infty}^\infty \mathrm{d}\mu \quad p(\mu) \frac{1}{\sqrt{2\pi}\sigma} \exp \left[-\frac{\left(\mu - x \right)^2}{2\sigma^2} \right] = f(x)$$

I'm just stumped on ideas for distribution functions. I'm not familiar with many besides gaussian, maxwellian etc. I've been trying to fit them with a linear combination of gaussian * polynomials, but I end up with a lot of oscillations and no good fit for the tail on the right hand-side.

Is anyone here great at looking at plots and knowing a good form for the distribution function?! Thanks in advance!

• Because there are myriad ways to express PDFs, a good solution should aim for a representation that is suitable for your purpose: what, then, is the objective of this exercise? – whuber Jun 23 '14 at 18:14
• Ok, the main thing is that I need to be able to integrate the pdf * a gaussian (I added the expression above) analytically, uniquely for each pdf. – cnosam Jun 23 '14 at 18:22
• Where do these PDFs come from and in what mathematical form are they currently represented? – whuber Jun 23 '14 at 18:37
• It's a probability of magnification, they're currently just histograms (these are gaussian kde plots). Physically I think there is motivation to use a log-normal distribution, but that's hard to convolve with the gaussian. – cnosam Jun 24 '14 at 0:28
• So what you are really saying is that you have batches of univariate data: sets of numbers, not distributions! That makes all the difference because it opens up many more options for the solution. Please consider editing your question to include an explanation of these data and of why you will need to perform the convolutions. – whuber Jun 24 '14 at 14:02

## 1 Answer

One way to do this is by using an L-moment diagram. You estimate the L-moments for each of the datasets for which you want to estimate the distribution, and then plot them on a L-moment diagram to identify the distribution type.

There are several papers in the hydrological literature about how to do this, and there is a dedicated R package (http://www.cran.r-project.org/web/packages/lmomco/index.html) to construct the L-moment diagram for sample data, together with the theoretical points, curves and regions that would help to identify the distribution type.

• yes (+1) but it seems the OP wants an automated solution. – user603 Jun 23 '14 at 20:19
• Ah, some clarification would be useful then. I read the comment from @cnosam "looking at plots and knowing a good form for the distribution function" and assumed this was a manual decision-making process. – hydrologist Jun 23 '14 at 20:30
• Thanks, but yes, it needs to pretty manual - they all need the same DF and it needs to convolve with the gaussian easily (see post) – cnosam Jun 24 '14 at 0:26
• OK - but it might be more feasible to start by characterising the distributions you have, and then using that to determine how to find $p\left(\mu\right)$. If you can find a distribution family that fits all of the curves, then you will have its PDF and will be able to see how this could be expressed as in your question - does that make sense? It would be useful to know how these PDFs are generated too. – hydrologist Jun 24 '14 at 0:35