One way to do this is to start with discrete distributions, then modify them by adding continuous noise to get continuous distributions, if continuous distributions are desired. The nice thing about discrete distributions is that it is very easy to manipulate them to get various values of skewness, kurtosis, etc.
The following code only deals with skewness and kurtosis. To change the standard deviation parameter, all that is needed is to multiply the data values by a scale factor. (For example, multiplying $x$ by 2 increases the standard deviation twofold.)
Here is code to calculate the skewness and kurtosis of discrete distributions whose values are in "x" and whose associated probabilities are in "p."
skew <-function(x,p) {
k = length(x)
m = sum(x*p)
v = sum( (x-m)^2 *p)
m3 = sum( (x-m)^3 *p)
sk = m3/v^1.5
return(sk)
}
kurt <-function(x,p) {
k = length(x)
m = sum(x*p)
v = sum( (x-m)^2 *p)
m4 = sum( (x-m)^4 *p)
k = m4/v^2
return(k)
}
With this code it is possible to generate all kinds of skewness and kurtosis values by playing with the "x" and "p." For example, a flat-topped leptokurtic distribution can be generated as follows:
#Example 1: Flat-topped leptokurtic distribution
x = c(1:4,10)
p = c(.24,.24,.24,.24,.04)
skew(x,p)
kurt(x,p)
plot(x,p, type="h", lwd=2, ylim = c(0, max(p)*1.2))
The skewness of this distribution is 2.24, the kurtosis is 9.80, and its graph is as follows:
If a data set is needed, you can sample from the distribution as follows:
set.seed(12345)
n=10000
x.sample = sample(x, n, replace=T, p)
If continuous data is needed you can jitter or add noise:
x.sample = x.sample + .2*rnorm(n)
The skewness, kurtosis, and distributional shape properties of the smoothed sample are similar to those of the discrete distribution, as shown by the following code:
library(moments)
skewness(x.sample)
kurtosis(x.sample)
hist(x.sample, breaks=30, main = "Flat-topped but Leptokurtic")
The sample skewness and kurtosis are 2.19 and 9.74, and the histogram looks as follows:
As another example, you can easily create an example of data that are "peaked" but platykurtic, as follows:
# Example 2: Peaked platykurtic distribution
x = 1:9
p = c(rep(.08,4), .36, rep(.08,4))
skew(x,p)
kurt(x,p)
plot(x,p, type="h", lwd=2, ylim = c(0, max(p)*1.2))
xs = sample(x, n, replace=T, p) + .2*rnorm(n)
skewness(xs)
kurtosis(xs)
hist(xs, breaks=30, main="Peaked but Platykurtic")
The skewness and kurtosis of the discrete distribution are 0 and 2.46 (<3 implies platykurtic), and the smoothed data sample have similar values. The histogram of the continuously smoothed data set illustrates the peakedness (despite being platykurtic) clearly:
A more difficult problem is to start with skewness and kurtosis values, and have the computer automatically select x and p to give those values. The optimization routines in R can help here, but there are difficulties in that there may be infinitely many solutions, or no solutions at all as whuber noted in a comment.
