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The logitnormal distribution is useful as a prior density for variables that are bounded between 0 and 1, such as proportions. The following figure displays its density for various combinations of parameters mu (panels) and sigma (lines).
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
plot of chunk densityPlots
Example: Plot the cumulative distribution
x <- seq(0,1, length.out=81)
d <- plogitnorm(x, mu=0.5, sigma=0.5)
plot(d~x,type="l")
plot of chunk cumDensityPlot
The moments have no analytical solution. This package estimates them by numerical integration:
Example: estimate mean and standard deviation.
(theta <- momentsLogitnorm(mu=0.6,sigma=0.5))
## mean var
## 0.63812093 0.01208171
The mode is found by setting derivatives to zero and optimizing the resulting equation: \(logit(x) = \sigma^2(2x-1)+\mu\).
Example: estimate the mode
(mle <- modeLogitnorm(mu=0.6,sigma=0.5))
## [1] 0.6641416
from upper quantile and
Example: estimate the parameters, with mode 0.7 and upper quantile 0.9
(theta <- twCoefLogitnormMLE(0.7,0.9))
## mu sigma
## [1,] 0.7608886 0.464783
x <- seq(0,1, length.out=81)
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7,0.9), col="grey")
plot of chunk twCoefLogitnormMLE
When increasing the \(\sigma\) parameter, the distribution becomes
eventually becomes bi-model, i.e. has two maxima. The unimodal distribution for
a given mode with widest confidence intervals is obtained by
function twCoefLogitnormMLEFlat
.
(theta <- twCoefLogitnormMLEFlat(0.7))
## mu sigma
## [1,] 0.01213214 1.444962
x <- seq(0,1, length.out=81)
d <- dlogitnorm(x, mu=theta[1,"mu"], sigma=theta[1,"sigma"])
plot(d~x,type="l")
abline(v=c(0.7), col="grey")
plot of chunk twCoefLogitnormMLEFlat
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.