betareg

Create a Zero- and/or One-Inflated Beta Distribution

Description

Class and methods for zero- and/or one-inflated beta distributions in regression specification using the workflow from the distributions3 package.

Usage

Beta01(mu, phi, p0 = 0, p1 = 0)

Arguments

`mu`	numeric. The mean of the beta distribution (on the open unit interval).
`phi`	numeric. The precision parameter of the beta distribution.
`p0`	numeric. The probability for an observation of zero (often referred to as zero inflation).
`p1`	numeric. The probability for an observation of one (often referred to as one inflation).

Details

The zero- and/or one-inflated beta distribution is obtained by adding point masses at zero and/or one to a standard beta distribution.

Note that the support of the standard beta distribution is the open unit interval where values of exactly zero or one cannot occur. Thus, the inflation jargon is rather misleading as there is no probability that could be inflated. It is rather a hurdle or two-part (or three-part) model.

Value

A Beta01 distribution object.

Examples

library("betareg")


## package and random seed
library("distributions3")
set.seed(6020)

## three beta distributions
X <- Beta01(
  mu  = c(0.25, 0.50, 0.75),
  phi = c(1, 1, 2),
  p0 = c(0.1, 0, 0),
  p1 = c(0, 0, 0.3)
)

X

[1] "Beta01 distribution (mu = 0.25, phi = 1, p0 = 0.1, p1 = 0.0)"
[2] "Beta01 distribution (mu = 0.50, phi = 1, p0 = 0.0, p1 = 0.0)"
[3] "Beta01 distribution (mu = 0.75, phi = 2, p0 = 0.0, p1 = 0.3)"

## compute moments of the distribution
mean(X)

[1] 0.225 0.500 0.825

variance(X)

[1] 0.090000 0.125000 0.056875

## support interval (minimum and maximum)
support(X)

     min max
[1,]   0   1
[2,]   0   1
[3,]   0   1

## simulate random variables
random(X, 5)

            r_1         r_2         r_3        r_4       r_5
[1,] 0.01770077 0.031967958 0.009185013 0.09774271 0.5297302
[2,] 0.11888790 0.004743561 0.209872794 0.56026234 0.7010201
[3,] 0.43181699 0.757365599 0.612582662 0.76991426 0.3122026

## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ])

hist(x[2, ])

hist(x[3, ])

## probability density function (PDF) and log-density (or log-likelihood)
x <- c(0.25, 0.5, 0.75)
pdf(X, x)

[1] 0.6156832 0.6366198 0.7718605

pdf(X, x, log = TRUE)

[1] -0.4850227 -0.4515827 -0.2589515

log_pdf(X, x)

[1] -0.4850227 -0.4515827 -0.2589515

## cumulative distribution function (CDF)
cdf(X, x)

[1] 0.6808374 0.5000000 0.2737016

## quantiles
quantile(X, 0.5)

[1] 0.05868041 0.50000000 0.94876688

## cdf() and quantile() are inverses
cdf(X, quantile(X, 0.5))

[1] 0.5 0.5 0.5

quantile(X, cdf(X, 1))

[1] 1 1 1

## point mass probabilities (if any) on boundary
cdf(X, 0, lower.tail = TRUE)

[1] 0.1 0.0 0.0

cdf(X, 1, lower.tail = FALSE)

[1] 0.0 0.0 0.3

## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)

         q_0.05      q_0.5    q_0.95
[1,] 0.00000000 0.05868041 0.8991438
[2,] 0.00615583 0.50000000 0.9938442
[3,] 0.28573175 0.94876688 1.0000000

quantile(X, p, elementwise = TRUE)

[1] 0.0 0.5 1.0

quantile(X, p, elementwise = TRUE, drop = FALSE)

     quantile
[1,]      0.0
[2,]      0.5
[3,]      1.0

## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
  "theoretical" = mean(X),
  "empirical" = rowMeans(random(X, 1000))
)

     theoretical empirical
[1,]       0.225 0.2275291
[2,]       0.500 0.5091324
[3,]       0.825 0.8103725