simulate data for different sample size with same parameter

Question

all,

I'm trying to simulate data for different sample size (let say n= 10,20, and 30). At the same time, i need the parameter (in my case are skewness and kurtosis) for the data for each n is the same. How can i do it in R.

I already tried do like this:

set.seed(100)
n<-10
true.skew<-0
true.kur<-3

library(PearsonDS)
e<-rpearson(n,moments=c(mean=0,variance=1, skewness=true.skew, kurtosis=true.kur))

library(moments)

skewness(e)
kurtosis(e)

and the result are:

>skewness(e)
[1] 0.2110712
> kurtosis(e)
[1] 1.950325
> 

then i increase the sample size to n = 20,

set.seed(100)
n1<-20
true.skew<-0
true.kur<-3

library(PearsonDS)
e1<-rpearson(n1,moments=c(mean=0,variance=1, skewness=true.skew, kurtosis=true.kur))

library(moments)

skewness(e1)
kurtosis(e1)

output:

skewness(e1)
[1] 1.311147
> kurtosis(e1)
[1] 5.517818

the problems is, i got the different value of skewness and kurtosis.

Anybody can help me to do this?

Thank you.

Answer 1

Your problem may be small sample sizes.

For example, if I use n <- 1000 and the same seed then I get

> skewness(e)
[1] -0.04122395
> kurtosis(e)
[1] 3.026162

which are a lot closer to the preset population statistics of $0$ and $3$

It is unreasonable to expect that for a single small sample the sample skewness and kurtosis will be close to the population figure. Indeed the sample kurtosis seems to be biased down for small samples. Try this code:

library(PearsonDS)
library(moments)
set.seed(100)
n <- 10
cases <- 1000
true.skew <- 0
true.kur <- 3

em <- matrix(rpearson(n*cases,
      moments=c(mean=0,variance=1, skewness=true.skew, kurtosis=true.kur)),
      ncol=cases)

sk <- apply(em,2,skewness)
ku <- apply(em,2,kurtosis)
plot(sk,ku) 

to give

and you would also get a smile or part of a smile sampling from other distributions.

If you want to compare different regression methods for small sample sizes, you should take many samples of the same size and see how the results vary.