How can one confidently choose a p and q for a beta distribution? Is there a methodology or R tool that allows you to best attribute a p and q value for a Beta distribution? I am currently building a simulation that involves stochastic processes, and from the little empirical data I was able to obtain (30 data points) I developed a Cullen and Frey graph with bootstrapping to understand better what distribution my data follows, in this case, Beta. However, I have to also provide a p (lower shape parameter > 0), and a q (upper shape parameter > 0) for the distribution. I understand what they do however, I would like to know if there is a standard process to help one identify or estimate these parameters better.
 A: There are a lot of ways to perform parameter estimation. This wikipedia page mentions a dozen of different types or related concepts:

Commonly used estimators (estimation methods) and topics related to them include:

*

*Maximum likelihood estimators

*Bayes estimators

*Method of moments estimators

*Cramér–Rao bound

*Least squares

*Minimum mean squared error (MMSE), also known as Bayes least squared error (BLSE)

*Maximum a posteriori (MAP)

*Minimum variance unbiased estimator (MVUE)

*Nonlinear system identification

*Best linear unbiased estimator (BLUE)

*Unbiased estimators — see estimator bias.

*Particle filter

*Markov chain Monte Carlo (MCMC)

*Kalman filter, and its various derivatives

*Wiener filter


As Christian Henning mentioned in the comments the beta distribution has some simple solutions for a few of those methods.

This shows that there are a lot of various ways to estimate the parameters. These have all different advantages and disadvantages. Whichever is the 'most optimal way' will depend on the context.
This is not a standard straightforward filling in a function in R-code as if it is some oracle that gives you an answer to any dataset. You have to think yourselves as well about what you want the function/algorithm (the "oracle") to do for you.
