The bridgedist Basics

Bruce J. Swihart


## The argument `scale` has been replaced with `phi` in this version.
## The top of page 77o in Wang Louis 2003 states phi is not a scale.
## Please start using the `phi` argument now.

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

xaxis = seq(-4,4,.01)
df = data.frame( xaxis,
                 Bridge = dbridge(xaxis, phi=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") 
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.

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 (1/phi).

phi <- 0.5
df = data.frame(
                 Bridge = rbridge(1e5, phi=phi),
                 Std_Logistic = rlogis(1e5),
                 BridgePlusStd_Logistic = rbridge(1e5, phi=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) +

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.