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")
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()
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