In JonasMoss/univariateML: Maximum Likelihood Estimation for Univariate Densities

knitr::opts_chunk$set(echo = TRUE)

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")

JonasMoss/univariateML documentation built on Feb. 6, 2024, 2:21 p.m.