library(bridgedist)

Recreate the plot from Wang and Louis (2003) where the Bridge, Normal, and Logistic all have unit variance and mean 0 with ggplot2:

library(reshape2)
library(ggplot2)
xaxis = seq(-4,4,.01)
df = data.frame( xaxis,
                 Bridge = dbridge(xaxis, scale=1/sqrt(1+3/pi^2)),
                 Normal = dnorm(xaxis),
                 Logistic = dlogis(xaxis, scale=sqrt(3/pi^2)))
melt.df <- melt(df, id.vars = "xaxis")
colnames(melt.df) <- c("x", "Distribution", "value")
ggplot(melt.df, aes(x, value, color=Distribution)) + 
  geom_line(size=1.05) + 
  ylab("Probability density function")
Fig. 1. Probability density functions of the Gaussian, logistic and bridge, for logistic, distributions each with zero mean and unit variance.

Fig. 1. Probability density functions of the Gaussian, logistic and bridge, for logistic, distributions each with zero mean and unit variance.

The implication is that a random variable from a Bridge distribution plus random variable from a standard logistic distribution is a logistic random variable with a scale greater than one.

phi <- 0.5
df = data.frame(
                 Bridge = rbridge(1e5, scale=phi),
                 Std_Logistic = rlogis(1e5),
                 BridgePlusStd_Logistic = rbridge(1e5, scale=phi) +  rlogis(1e5),
                 Logistic = rlogis(1e5, scale=1/phi)
)
melt.df <- melt(df)
colnames(melt.df) <- c("Distribution", "value")
ggplot(melt.df, aes(value)) +
  facet_grid(.~Distribution) +
  geom_histogram()
Fig. 2. 10000 random variates in each panel. From left to right: the bridge distribution, the logistic with scale=1, the sum of the previous two, and the logistic with scale=1/phi. Note how similar the third and fourth panel, the application supporting the theory.

Fig. 2. 10000 random variates in each panel. From left to right: the bridge distribution, the logistic with scale=1, the sum of the previous two, and the logistic with scale=1/phi. Note how similar the third and fourth panel, the application supporting the theory.