Asymetric and mesokurtic distribution?

Question

I need to find an example of an asymetric distribution with similar kurtosis to the normal distribution, and generate a sample in R.

I've already chosen the 4 density of Marron and Wand as an example of a symetric distribution with different kurtosis than the normal distribution, but I coudn't find the former.

Thank you!

Answer 1

Using the example from Wikipedia (mentioned in a Comment), here is a simulation of a million observations of $Y = (X - E(X))/SD(X),$ where $X \sim \mathsf{Binom}(n = 10, p = .5+\sqrt{1/12}).$ By standardizing $X,$ we make the kurtosis of $Y$ be $E(Y^4) = 3,$ which matches the kurtosis of a normal distribution.

Binomial distributions are easy to simulate. In R statistical software the function rbinom samples from a binomial distribution according to designated parameters. With a million iterations one can expect two digits of accuracy.

set.seed(413)
m = 10^6;  n = 10;  p = .5 + sqrt(1/12)
x = (rbinom(m, n, p) - n*p)/sqrt(n*p*(1-p))
mean(x);  mean(x^4)
## -0.001070761    # aprx E(X) = 0
## 2.989588        # aprx E(X^4) = 3

I will leave it to you to verify analytically, perhaps using moment generating functions, that Wikipedia's assertion about the kurtosis of this binomial distribution is precisely correct.