# features of probability distributions

What kind of probability distribution features one can make use of in order to identify a specific pdf? I was thinking of using the moments of the distribution (mean,var,skewness and kurtosis) but I am not sure whether specific moments (or combination of them) identify specific pdf.

Edited : The problem is the following:

I would like to know which features do I need to extract from a pdf in order to use some ML techniques and train a model such that it may perform a pdf clustering, i.e. train a model such that it can be able to classify whether a new data set (random values) may be represented by a certain pdf or not. Basically, something similar to GOF tests but using some more sophisticated techniques ( if they exist ).

• Features alone won't do it for you. – Michael R. Chernick Mar 29 '17 at 15:29
• It depends on how you are going to use the results and on the data available to you. Why don't you explain the particular problem you are trying to address? – whuber Mar 29 '17 at 15:32
• If you knew all moments that would define the distribution, but in empirical situations you don't know all of them. You only know all moments when working with some kind of a model – Aksakal Mar 29 '17 at 15:36

A related idea is used in the context of Approximate Bayesian Computation, where people have an original data set, simulate subsequent data sets from the model, and check whether they are "similar" to the original one. This is done by using: (i) a set of summary statistics $S$ (such as moments, quantiles, and etcetera), (ii) a distance between the summary statistics $\delta$, and (iii) a tolerance $\epsilon>0$.
This, is, $Data_2$ is accepted as an approximate "replication" (in a loose sense) of $Data_1$ if $\delta(S(Data_1),S(Data_2))<\epsilon$. There is no unique to choose $\delta$, $S$, and $\epsilon$.