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

density functiondescriptive statisticsdistributionskurtosisskewness

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.

Best Answer

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.

Related Solutions

Solved – How to create a random variables in a simulation using skewness and kurtosis as well as average and standard deviation input

These features can be included in simulations from a symmetric distribution using transformations that control skewness and kurtosis such as the Johnson-SU transformation, the g-and-h, the g-and-k, the sinh-arcsinh, and the LambertW tranformations (Tukey-type transformations). A quick google search gives you relevant references for these transformations. See also Transformation to increase kurtosis and skewness of normal r.v

Solved – Skewness of a mixture density

Skewness is a vague concept which allows its formalisation in several ways. The most popular measure of skewness is the one you mention, which was proposed more than 100 years ago. However, there are better (more interpretable) measures nowdays.

It has been largely discussed the validity of using a measure of skewness in multimodal distributions, since its interpretation becomes unclear. This is the case of finite mixtures. If your mixture looks (or is) unimodal, then you can use this value to understand a bit how asymmetric it is.

In R, this quantity is implemented in the library moments, in the command skewness().

The moments of a mixture $X$, with density $g=\sum_{j=1}^n \pi_j f_j$ can be calculated as $E[X^k] = \sum_{j=1}^n \pi_j\int x^k f_j(x)dx$.

A numerical solution in R.

# Sampling from a 2-gaussian mixture
gaussmix <- function(n,m1,m2,s1,s2,alpha) {
    I <- runif(n)<alpha
    rnorm(n,mean=ifelse(I,m1,m2),sd=ifelse(I,s1,s2))
}

# A simulated sample
samp <- gaussmix(100000,0,0,1,1,0.5)

library(moments)
# Approximated kurtosis and skeweness using the simulated sample
skewness(samp)
kurtosis(samp)

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Solved – How to create a random variables in a simulation using skewness and kurtosis as well as average and standard deviation input

Solved – Skewness of a mixture density

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