Given a probability density function supported on a semi-infinite interval obtained numerically. What is the best methodology to follow in order to know / estimate the distribution law (Beta prime, chi, gamma, log-normal, etc.)
Enclosed is an example of a PDF obtained numerically.
It is supposed that we know the exact distribution law? If so, then it is a matter of
1)either a Least Squares Estimation (LSE)
2)or a Maximum Likelihood Estimation (MLE)
If we are interested only for the critical points of the distribution, then there exist other non parametrical methods for doing this.
I shall come back again.
It is supposed that we know the exact distribution law? If so, then it is a matter of
1)either a Least Squares Estimation (LSE)
2)or a Maximum Likelihood Estimation (MLE)
If we are interested only for the critical points of the distribution, then there exist other non parametrical methods for doing this.
I shall come back again.
You can also use a non parametric approach with Kernel Density Estimation (KDE) techniques.
More details about KDE can be found here:
Härdle, W. et al., Nonparametric and semiparametric models - an introduction, 344 pp., 2005. http://www.getiezewdie.com/wp-content/uploads/2013/05/Nonparametric-and-Semiparametric-Models-An-introduction.pdf
You can use for example the excellent KDE toolbox for Matlab(r), for example.
Sorry, the website of the KDE toolbox is missing:
http://www.ics.uci.edu/~ihler/code/kde.html
As Demetris has rightly pointed out you can try and fit a named few distributions that you think matches its shape and select the best one
First, find out the moments numerically, then decide yourself from the existing distributions that suitably fits your data. or you may workout with the empirical distribution function. The other option is to develop a function with the help of nonlinear least squares method.
Caution about moments: There exist not so many algorithms to ensure that they are finite.
The method you use likely depends on what you know or are willing to assume about the underlying process. The distribution you have provided looks remarkably like a Gamma distribution with a shape parameter =3 and scale parameter =1. If you know that the underlying process is Gamma (or are willing to accept an approximation to your process) then estimating the particular Gamma, via, for example, maximum likelhood is a very efficient route to an estimated density. If you do not wish to make the assumption of a form, such as Gamma, or some richer parametric family of defined distributions with densities, then you are led to nonparametric density estimation, as mentioned by Phillipe Blanc above. The Härdle book is good, but the literature is vast. It is worth noting that:
It takes a lot more data to estimate a density from scratch--that is making NO assumptions about form.
This performace gap can be reduced by making assumptions on the smoothness of your underlying density: differentiable, twice differentiable, etc.
Most statistical packages have routines for fitting both parametric distributions and nonparametric densities, so you have ready tools to apply to your data.
As Balázs Kotosz notes, you can go all the way to methods that derive from the general problem of approximating functions.
I've attached a pdf of a histogram based on 10,000 simulated gamma(3,1) variates, with a mximum likelihood fit of a gamma distribution and a kernel density fit. Note that the kernel density is still following the observational 'bumps' of the simulation.
Thank you for you all for your powerful discussions! They indeed gave me a nice idea about how to proceed in order to get the desired PDF. Thank you Mike for giving a starting point for my distribution. In fact, I have a feeling that this distribution is very similar of the Richardson PDF used in particle dispersion in turbulence.
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