These are the currently implemented distributions.

Name univariateML function Package Parameters Support
Cauchy distribution mlcauchy stats location,scale \(\mathbb{R}\)
Gumbel distribution mlgumbel extraDistr mu, sigma \(\mathbb{R}\)
Laplace distribution mllaplace extraDistr mu, sigma \(\mathbb{R}\)
Logistic distribution mllogis stats location,scale \(\mathbb{R}\)
Normal distribution mlnorm stats mean, sd \(\mathbb{R}\)
Beta prime distribution mlbetapr extraDistr shape1, shape2 \((0, \infty)\)
Exponential distribution mlexp stats rate \([0, \infty)\)
Gamma distribution mlgamma stats shape,rate \((0, \infty)\)
Inverse gamma distribution mlinvgamma extraDistr alpha, beta \((0, \infty)\)
Inverse Gaussian distribution mlinvgauss actuar mean, shape \((0, \infty)\)
Inverse Weibull distribution mlinvweibull actuar shape, rate \((0, \infty)\)
Log-logistic distribution mlllogis actuar shape, rate \((0, \infty)\)
Log-normal distribution mllnorm stats meanlog, sdlog \((0, \infty)\)
Lomax distribution mllomax extraDistr lambda, kappa \([0, \infty)\)
Rayleigh distribution mlrayleigh extraDistr sigma \([0, \infty)\)
Weibull distribution mlweibull stats shape,scale \((0, \infty)\)
Log-gamma distribution mllgamma actuar shapelog, ratelog \((1, \infty)\)
Pareto distribution mlpareto extraDistr a, b \([b, \infty)\)
Beta distribution mlbeta stats shape1,shape2 \((0, 1)\)
Kumaraswamy distribution mlkumar extraDistr a, b \((0, 1)\)
Logit-normal mllogitnorm logitnorm mu, sigma \((0, 1)\)
Uniform distribution mlunif stats min, max \([\min, \max]\)
Power distribution mlpower extraDistr alpha, beta \([0, a)\)

This package follows a naming convention for the ml*** functions. To access the documentation of the distribution associated with an ml*** function, write package::d***. For instance, to find the documentation for the log-gamma distribution write

?actuar::dlgamma

Problematic Distributions

Lomax Distribution

The maximum likelihood estimator of the Lomax distribution frequently fails to exist. For assume \(\kappa\to\lambda^{-1}\overline{x}^{-1}\) and \(\lambda\to0\). The density \(\lambda\kappa\left(1+\lambda x\right)^{-\left(\kappa+1\right)}\) is approximately equal to \(\lambda\kappa\left(1+\lambda x\right)^{-\left(\lambda^{-1}\overline{x}^{-1}+1\right)}\) when \(\lambda\) is small enough. Since \(\lambda\kappa\left(1+\lambda x\right)^{-\left(\lambda^{-1}\overline{x}^{-1}+1\right)}\to\overline{x}^{-1}e^{-\overline{x}^{-1}x}\), the density converges to an exponential density.

eps = 0.1
x = seq(0, 3, length.out = 100)
plot(dexp, 0, 3, xlab = "x", ylab = "Density", main = "Exponential and Lomax")
lines(x, extraDistr::dlomax(x, lambda = eps, kappa = 1/eps), col = "red")

plot of chunk lomax