I don't think you need a separate post. This thread is fine.
As I have already told you, it is impossible to have a probability distribution for which kurt=3 when skew>1.5. You can only use feasible pairs of (skew,kurt) values. In general, the impossible region is defined by kurt >= 1 + skew**2. However, the Fleishman family cannot model the most extreme distributions. Here is some DATA step code to get only the feasible pairs that can be fit by the Fleishman family:
/* create (skew,kurt) values for skew > 0 that can be fit by Fleishman family */
data FeasSkewKurt;
do skew = 0 to 2.4 by 0.2;
do kurt = -2 to 10 by 0.5;
/* keep only valid pairs */
if kurt > (-1.2264489 + 1.6410373* skew**2) then output;
end;
end;
run;
The main question you need to answer is WHAT DISTRIBUTIONS do you want to simulate from? You originally said beta distributions, which are bounded. You can either choose from standard families (such as beta, gamma, lognormal,...) and try to get a wide range of (skew,kurt) values, or you can use a flexible family of distributions such as the Fleishman family or the Johnson system. Using a family such as truncated normals or a mixture of normals is going to greatly complicate your life, so I do not recommend using those families. (The problem is that it is hard to find parameter values for each (skew,kurt) pair when you use those distributions.)
The basic idea of what you are trying to do is discussed and implemented in Simulating Data with SAS (Wicklin, 2013) in Chapter 16 "Moment Matching and the Moment-Ratio Diagram." In that chapter, I used the Fleishman family, but the same ideas apply to the Johnson system. I recommend either of those families. If you do not have access to that book, you can get the Fleishman functions for free from Appendix D, which is available at https://support.sas.com/en/books/authors/rick-wicklin.html You are comfortable using SAS/IML to simulate the data, you could then write the samples to a data set and use the simulated data anywhere in SAS.
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