Ozibeta: The Zero/One-Inflated Beta Distribution

Description

Density, distribution function, and random generation for the zero/one-inflated beta distribution.

Usage

dozibeta(x, shape1, shape2, pobs0 = 0, pobs1 = 0, log = FALSE,
         tol = .Machine$double.eps)
pozibeta(q, shape1, shape2, pobs0 = 0, pobs1 = 0,
         lower.tail = TRUE, log.p = FALSE, tol = .Machine$double.eps)
qozibeta(p, shape1, shape2, pobs0 = 0, pobs1 = 0,
         lower.tail = TRUE, log.p = FALSE, tol = .Machine$double.eps)
rozibeta(n, shape1, shape2, pobs0 = 0, pobs1 = 0,
         tol = .Machine$double.eps)

Arguments

x, q, p, n

Same as Beta.

pobs0

vector of probabilities that 0 are observed ($\omega_0$).

pobs1

vector of probabilities that 1 are observed ($\omega_1$).

shape1, shape2

Same as Beta. They are called a and b in beta respectively.

lower.tail, log, log.p

Same as Beta.

tol

Numeric, tolerance for testing equality with 0.

Value

dozibeta gives the density, pozibeta gives the distribution function, qozibeta gives the quantile, and rozibeta generates random deviates.

Details

This distribution is a mixture of a discrete distribution with a continuous distribution. The cumulative distribution function of $Y$ is

F (y) = (1 - ω_{0} - ω_{1}) B (y) + ω_{0} \times I [0 \leq y] + ω_{1} \times I [1 \leq y]

where $B(y)$ is the cumulative distribution function of the beta distribution with the same shape parameters (pbeta), $\omega_0$ is the inflated probability at 0 and $\omega_1$ is the inflated probability at 1. The default values of $\omega_j$ mean that these functions behave like the ordinary Beta when only the essential arguments are inputted.

Examples

Run this code

set.seed(208); N <- 10000
k <- rozibeta(N, 2, 3, 0.2, 0.2)
hist(k, probability = TRUE, border = "blue",
     main = "Blue = inflated; orange = ordinary beta")
sum(k == 0) / N  # Proportion of 0
sum(k == 1) / N  # Proportion of 1
Ngrid <- 1000
lines(seq(0, 1, length = Ngrid),
      dbeta(seq(0, 1, length = Ngrid), 2, 3), col = "orange")
lines(seq(0, 1, length = Ngrid), col = "blue",
      dozibeta(seq(0, 1, length = Ngrid), 2 , 3, 0.2, 0.2))

set.seed(1234); k <- runif(1000)
sum(abs(qozibeta(k,  2, 3) - qbeta(k, 2,  3)) > .Machine$double.eps)  # Should be 0
sum(abs(pozibeta(k, 10, 7) - pbeta(k, 10, 7)) > .Machine$double.eps)  # Should be 0

Run the code above in your browser using DataLab

Last chance! 50% off unlimited learning