Distributions

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")