For a given random variable, I have performed a series of calculations and noticed that the kurtosis evolves linearly with respect to the square of skewness. My purpose here is to find out a two parameter probability density function that fits well my data obtained numerically. Is there any common method to use in order to get the desired distribution? (I have put in attach the evolution law). Thanks!
Hello,
It seems you have two separate problems here. One is of a more mathematical nature, namely to determine whether one can find a PDF for which skewness is A*kurtosis, for any A. I suppose you can work with MGFs and try to solve for f(X), but I don't see any easy way of doing that. If I find some spare time I may play with the maths behind that question.
But your problem seems really to be which location-scale distribution would best fit your data. There are hundreds of thousands of ways of doing that. You can use maximum likelihood- or moments-based methods.
Take a look at the links below and see if they are of any help:http://lamages.blogspot.com.br/2011/12/fitting-distributions-with-r.html
http://stat.ethz.ch/R-manual/R-patched/library/MASS/html/fitdistr.html
http://cran.r-project.org/web/packages/fitdistrplus/fitdistrplus.pdf
Cheers,
Luiz
Also, take a look here. It seems physicists have already found K = AS +B
Take a look here
http://xa.yimg.com/kq/groups/13354653/1407977864/name/b.pdf
Cheers,
Luiz
No, you cannot.
Skewness and Kurtosis deal with the amplitude distribution of your samples only while the spectrum depends also on the timing between them. If you order your samples monotonically they will have the same K and S, but the PDF would have changed.
See
www.supermathematica.com; www.supermatematica.ro;
www.supermatematicaonline.blogspot.ro
Kurtosis and skewness helps in understanding the nature of distribution of values assumed by variable while the PDF indicates the values assumed with certain probability value for a variable. From PDF the skewness and kurtosis may be easily determined.
This problem cannot be solved from the mathematical point of view. If you open any book related to the problem of moments you can meet the theorem which was considered in the rather old book "The Advanced Theory of Statistics" v.1. written by M. Kendall ans A. Stuart 50 years ago. They considered in details the so-called problem of the moments and proved that any integer set of constants satisfying to certain conditions cannot restore the desired PDF. But if you consider the set of moments including the fractional ones the problem can be solved! Partly I considered this problem in my publication related to presentation of a wide set of random functions with unknown PDF in the space of moments. I'm enclosing this publication for your info. Recently with my italian collegues we submitted more powerful approach which was received the abbreviation FERMA- Fractional Exponential Reduced Moments Approach.
If you know the first four moments of the distribution you can use these to determine a distribution that in some sense is the best under these constranits. The most common method to use the first terms in the Edgeworth expansion. This method is simple but far large deviations from the mean it often gives misleading result such as negative values of the probability density function. To avoid negative probabilities one can instead use a Cornish-Fisher expansion that automatically gives positive values, but there are much fewer references to this method. Finally one can calculate the maximum entropy distribution under the given constraints. If only the first three moments are given no maximum entropy distribution exists (except if the skewness is zero) but if four moments are given a maximum entropy distribution normally exists (see my paper Lower Bounds for Divergence in the Central Limit Theorem for details).
No. The space of probability density functions in infinite dimensional. The space of the pair given by kurtosis and skewness is two dimensional. Hence there is no 1-1 mapping.
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