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 $$\mathbb{R}$$
Student t distribution mlstd fGarch mean, sd, nu $$\mathbb{R}$$
Generalized Error distribution mlged fGarch mean, sd, nu $$\mathbb{R}$$
Skew Normal distribution mlsnorm fGarch mean, sd, xi $$\mathbb{R}$$
Skew Student t distribution mlsstd fGarch mean, sd, nu, xi $$\mathbb{R}$$
Skew Generalized Error distribution mlsged fGarch mean, sd, nu, xi $$\mathbb{R}$$
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")