univariateML: Maximum Likelihood Estimation for Univariate Densities

title: 'univariateML: An R package for maximum likelihood estimation of univariate densities' tags: - R - statistics - maximum likelihood - density estimation authors: - name: Jonas Moss orcid: 0000-0002-6876-6964 affiliation: 1 affiliations: - name: University of Oslo index: 1 output: github_document: html_preview: true bibliography: paper.bib date: 25 November 2019

Summary

univariateML is an R [@r] package for user-friendly univariate maximum likelihood estimation [@lecam1990ml]. It supports more than 20 densities, the most popular generic functions such as plot, AIC, and confint, and a simple parametric bootstrap [@efron1994introduction] interface.

When looking at univariate data it is natural to ask if there is a known parametric density that fits the data well. The following example uses the egypt [@pearson1902egypt] data set included in the package and a plot of the Weibull and Gamma densities [@johnson1970continuous, Chapter 17 & 21].

# install.packages("univariateML")
library("univariateML")
hist(egypt$age, freq = FALSE, main = "Mortality", xlab = "Mortality")
lines(mlweibull(egypt$age)) # Plots a Weibull fit.
lines(mlgamma(egypt$age), col = "red")  # Plots a Gamma fit.

A natural question to ask is which among several models fits the data best. This can be done using tools of model selection such as the AIC [@akaike1998information].

AIC(mlweibull(egypt$age),
    mlgamma(egypt$age))

##                      df      AIC
## mlweibull(egypt$age)  2 1230.229
## mlgamma(egypt$age)    2 1234.772

Problems involving estimation of univariate densities are common in statistics. Estimation of univariate densities is used in for instance exploratory data analysis, in the estimation of copulas [@ko2019focused], as parametric starts in density estimation [@hjort_glad_1995; @moss2019kdensity], and is of interest in and of itself.

Analytic formulas for the maximum likelihood estimates are used whenever they exist. Most estimators without analytic solutions have a custom made Newton-Raphson solver. This is in contrast to the mle function in the built-in R package stats4, which supports more general maximum likelihood estimation through numerical optimization on a supplied negative log-likelihood function.

Rfast [@Rfast] is an R package with fast Newton-Raphson implementations of many univariate density estimators. univariateML differs from Rfast mainly in focus: While univariateMLis focused on user-friendly univariate density estimation, Rfast aims to have the fastest possible implementations of many kinds of functions.