# How do I reproduce this distribution (with observed means, sd, kurtosis, skewness and correlation)?

Let's move right onto the reproducible example using R:

library(moments)
library(magrittr)

dat <- data.frame(V1=c(1.36,0.65,0.66,1.27,0.57,0.58,0.85,0.67,1.56,1.28,0.59,0.64,0.8,0.8,0.74,1.01,0.68,0.6,0.5,1.56,0.76,0.96,1.16,0.73,0.61,0.67,0.82,0.69,0.5,0.47,0.68,1.34,0.97,0.5,1.64,1.12),
V2=c(173.6429,491.062,490.0896,190.3295,196.7952,147.3794,149.5065,209.2774,253.2763,204.3357,169.8099,196.8138,196.2807,198.8211,86.70567,118.3987,75.79,63.53013,388.96,182.9038,148.5611,192.9606,202.5833,164.6806,196.5939,114.2005,250.835,85.50414,118.0771,120.7658,486.2,246.1387,280.67,87.516,150.6643,280.5779),
V3=c(139,58,48,179,27,50,97,57,170,181,42,151,65,87,63,109,85,51,23,162,53,109,146,43,46,49,64,52,26,16,32,170,80,19,188,105))

n <- 1000

L_cor_mat <- cor(dat) %>% chol
apply(dat, 2, function(x){
sinh(kurtosis(x)*(asinh(rnorm(n, mean(x),sd(x)))+skewness(x)))
}) -> sim_var
(t(L_cor_mat) %*% t(sim_var)) %>% t -> sim_var

mean(dat$V1); mean(sim_var[,1]) sd(dat$V1); sd(sim_var[,1])
kurtosis(dat$V1); kurtosis(sim_var[,1]) skewness(dat$V1); skewness(sim_var[,1])

mean(dat$V2); mean(sim_var[,2]) sd(dat$V2); sd(sim_var[,2])
kurtosis(dat$V2); kurtosis(sim_var[,2]) skewness(dat$V2); skewness(sim_var[,2])

mean(dat$V3); mean(sim_var[,3]) sd(dat$V3); sd(sim_var[,3])
kurtosis(dat$V3); kurtosis(sim_var[,3]) skewness(dat$V3); skewness(sim_var[,3])

cor(dat); cor(sim_var)


Simply speaking, I want to reproduce V1, V2 and V3 in dat with larger sample sizes. I don't really know what parameters / distributions should I assume, but since there variables are highly correlated and skewed, I think I might have to consider altogether:

1. Mean
2. SD
3. Correlation matrix
4. Kurtosis
5. Skewness

Previously I was able to reproduce correlated normal variables (#1-3) with the instructions here: Generate Correlated Normal Random Variables

This time, I am referring to here for simulation with skewness and kurtosis: Transformation to increase kurtosis and skewness of normal r.v

But no more luck this time when I added in kurtosis and skewness (#4-5)... as you can see from the outputs of the above code, the outcomes do not match with the observed parameter values.

I am not really familiar with all those mathematics. Hopefully someone can guide me with the simulation. Thanks!

• "Simply speaking, I want to reproduce V1, V2 and V3 in dat with larger sample sizes." Why do you want this? What is your actual goal? Maybe you could simply bootstrap? – Roland Feb 14 '18 at 7:05
• @Roland Thanks for your response. I agree that bootstrapping can be a compromise in my case. Should have thought of that. However, I also feel a bit "uncomfortable" about the gap between samples during bootstrapping. Even though the true distribution is unknown, the variables should be well approximated by the criteria above (#1-5). So, if possible, it seems preferable to simulate a distribution over doing bootstrapping. This is not a must here, though (just trying to learn more, like when I learnt to simulate correlated multivariates). – Matthew Hui Feb 14 '18 at 7:25