betareg

Create a 4-Parameter Beta Distribution

Description

Class and methods for 4-parameter beta distributions in regression specification using the workflow from the distributions3 package.

Usage

Beta4(mu, phi, theta1 = 0, theta2 = 1 - theta1)

Arguments

`mu`	numeric. The mean of the beta distribution that is extended to support [theta1, theta2].
`phi`	numeric. The precision parameter of the beta distribution that is extended to support [theta1, theta2].
`theta1`, `theta2`	numeric. The minimum and maximum, respectively, of the 4-parameter beta distribution. By default a symmetric support is chosen by `theta2 = 1 - theta1` which reduces to the classic beta distribution because of the default `theta1 = 0`.

Details

The distribution is obtained by a linear transformation of a beta-distributed random variable with intercept theta1 and slope theta2 - theta1.

Value

A Beta4 distribution object.

Examples

library("betareg")


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

## three beta distributions
X <- Beta4(
  mu  = c(0.25, 0.50, 0.75),
  phi = c(1, 1, 2),
  theta1 = c(0, -0.1, -0.1),
  theta2 = c(1, 1.1, 1.5)
)

X

[1] "Beta4(mu = 0.25, phi = 1, theta1 =  0.0, theta2 = 1.0)"
[2] "Beta4(mu = 0.50, phi = 1, theta1 = -0.1, theta2 = 1.1)"
[3] "Beta4(mu = 0.75, phi = 2, theta1 = -0.1, theta2 = 1.5)"

## compute moments of the distribution
mean(X)

[1] 0.25 0.50 1.10

variance(X)

[1] 0.09375 0.18000 0.16000

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

      min max
[1,]  0.0 1.0
[2,] -0.1 1.1
[3,] -0.1 1.5

## simulate random variables
random(X, 5)

           r_1       r_2         r_3        r_4         r_5
[1,] 0.7497152 0.8385523  0.03196796 0.91882879  0.54543668
[2,] 0.1263742 1.0978772 -0.09430773 0.07151503 -0.00499443
[3,] 1.4311350 1.4850582  1.11178496 1.36985916  1.24450679

## 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] 0.6840925 0.5305165 0.4235834

pdf(X, x, log = TRUE)

[1] -0.3796622 -0.6339043 -0.8590048

log_pdf(X, x)

[1] -0.3796622 -0.6339043 -0.8590048

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

[1] 0.6453748 0.5000000 0.2022198

## quantiles
quantile(X, 0.5)

[1] 0.09331223 0.50000000 1.23888962

## 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

## 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,]  9.512588e-06 0.09331223 0.9118445
[2,] -9.261300e-02 0.50000000 1.0926130
[3,]  2.656317e-01 1.23888962 1.4975313

quantile(X, p, elementwise = TRUE)

[1] 9.512588e-06 5.000000e-01 1.497531e+00

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

         quantile
[1,] 9.512588e-06
[2,] 5.000000e-01
[3,] 1.497531e+00

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

     theoretical empirical
[1,]        0.25 0.2464581
[2,]        0.50 0.4930360
[3,]        1.10 1.1068752