Cullen and Frey graph can't recognize exponential distribution

Question

I am trying to familiarize myself with the Cullen and Frey plot. So, I generated 1000 exponentially distributed numbers using the command:

x <- rexp(1000, 2)

Then, with the fitdistrplus package, I created the Cullen and Frey plot.

install.packages("fitdistrplus")
library(fitdistrplus)
descdist(x, discrete = FALSE, boot = 200)

The problem is that it does not recognize the numbers in the x vector as exponentially distributed.

Why does this happen?

Answer 1

It so happens that skewness and kurtosis are not very accurately estimated for an exponential distribution, even with 1000 observations:

sk <- rep(10,10000)
for (i in seq_along(sk)) {
  x <- rexp(1000,2)
  sk[i] <- skewness(x)
}
quantile(sk^2, c(0.1,0.25,0.5,0.75,0.9))

10%      25%      50%      75%      90% 
2.893270 3.270011 3.786337 4.452599 5.262877 

We expect to see skewnesses less than 3.27 roughly 25% of the time.

Kurtosis is, as one might expect, no better; the code revision is straightforward so I don't present it here.

> quantile(ku, c(0.1,0.25,0.5,0.75,0.9))
      10%       25%       50%       75%       90% 
 6.426070  7.162297  8.187794  9.640750 11.569109 

Let's put this together and show, in a heuristic way, how unusual your results were for a sample size of 1000:

dt <- data.table(sk2 = rep(0, 1000), ku = rep(0, 1000))
for (i in 1:1000) {
  x <- rexp(1000,2)
  dt$sk2[i] <- skewness(x)^2
  dt$ku[i] <- kurtosis(x)
}
x <- densCols(dt$sk2, dt$ku, colramp=colorRampPalette(c("black", "white")))
dt$dens <- col2rgb(x)[1,] + 1L

## Map densities to colors
cols <-  colorRampPalette(c("#FF3100", "#FF9400", "#FCFF00", 
                            "#45FE4F", "#00FEFF", "#000099"))(6)
dt$col <- ifelse(dt$dens >= 250, cols[1], 
                 ifelse(dt$dens >= 200, cols[2], 
                        ifelse(dt$dens >= 150, cols[3], 
                               ifelse(dt$dens >= 100, cols[4], 
                                      ifelse(dt$dens >= 50, cols[5], cols[6])))))

## Plot it, reordering rows so that densest points are plotted on top
plot(ku~sk2, data=dt[order(dt$dens),], pch=20, col=col, cex=1, ylim=c(5,15), xlim=c(2,7))

biv_kde <- MASS::kde2d(x1, x2)
contour(biv_kde, add = T)

points(3, 6.8, pch=17, col=1)

... so your data, which is marked by the black triangle just below and to the left of the highest density region, doesn't appear to be too extreme.