PDF Formula for distribution with mean, standard deviation, skew, and kurtosis

Question

What would the probability density function be for a graph with input variables: mean, standard deviation, skewness, and kurtosis?

For example, if the inputs were confined only to mean and standard deviation, the formula would be:

$${\displaystyle f(x,\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$$

(the pdf formula for a normal distribution)

I looked on https://en.wikipedia.org/wiki/Pearson_distribution and found this:

$${\displaystyle p(x)={\frac {|{\frac {\Gamma(m+{\frac {\nu}{2}}i)}{\Gamma(m)}}|^{2}}{\alpha B(m-{\frac {1}{2}},{\frac {1}{2}})}}*[1+({\frac {x-\lambda}{\alpha}})^{2}]^{-m}*e^{-\nu *arctan({\frac {x-\lambda}{\alpha}})}}$$

It seems like it could be what I'm looking for, but I am unsure as to what all the symbols mean. If someone could explain, that would be great.

Answer 1

For a distribution with parameters for the mean, sigma, skewness, and kurtosis, I can only think of the Stable (Levy) distribution with its 4 parameters (mean=$\mu$, s.d.=$c$, skewness=$\beta$, and kurtosis=$\alpha$). The stable distribution can represent normal, skew-normal, logistic, Rayleigh, Cauchy, etc.

Answer 2

There are several families of distributions that can account for a variety of skewness and kurtosis levels. A good summary is

On Families of Distributions with Shape Parameters

These include:

1. Skew-Symmetric distributions
2. Two piece distributions
3. Sinh-arcsinh distributions.
4. Alpha-stable family
5. Many other families of distributions obtained by transforming the pdf/cdf or using parametric changes of variables.