# Distributions" In 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")


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univariateML documentation built on Jan. 25, 2022, 5:09 p.m.