Dear Ramin,

I offer you a variety of reference trough  links and attached files in topics.

-Mixture Models - CMU Statistics

https://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch20.pdf

-Binomial Gaussian mixture filter | EURASIP Journal on Advances in ...

https://asp.eurasipjournals.springeropen.com/.../s13634-015-0221-... 

Best regards

@ Ramin Esmzad

Studying the rich literature suggested by Mohamed-Mourad Lafifi should lead you to full knowledge about your problem. However, sometimes a little advice could help you to catch what specifically  you are  looking for,  already at the beginning. Therefore, please present more closely  the source of your problem. For instance, if the given p.d. is already a known mixture, then your problem is just an analysis of functions. In particular no statistical  procedures are required by the solution. A different case is, if you have data which are suspected of being from a population where the measured feature is a mixture (say of gaussian pd.-s) possible to be replaced by a single Gaussian. But then it depends on how strong is the convincement  that the p.d. is not a single Gaussian. Then a rigorously justified test  is required. This makes the problem more "advanced" ( Prof. Lafifi's reference list is definitly good source of such algorithms). Let me stress, however, that not every situation is described in books and papers. It is obvious, that some practice is required to decide what  tests should be performed to get an approprite decision. For instance, if the admitted hypothetic p.d.-s are close to a simple Gaussian, then the parametric test requires the choice of admitted number of parameters, or just to admit any p.d. and verify  the test by, say Kolmogov test. On the other hand if you are convinced that the number of components is at most 2, then a suitable parametric test can be used (probably you can find it in the literature, see e.g  from the paper Efficiency Learning. . .). So please supply a more detailed formulation of your problem. This should result in increasing a chance for giving you a more detailed answers (if they are expected, of course).

Best regards

Consider for example the Probabilistic Data Association Filter in which sum of Gaussian probability density functions are approximated by a Single Gaussian. My problem is exactly the same and I need more accurate approximations.

Thank you

@ Ramin Esmzad

Dear sir, please explain in your own words:

-- Is the approximation by PDAF  not the best?

-- What do you approxiamte? ( the given p.d. or data )

Best regards