VGAM (version 1.0-1)

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 $$F(y) =(1 - \omega_0 -\omega_1) B(y) + \omega_0 \times I[0 \leq y] + \omega_1 \times I[1 \leq y]$$ 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.

See Also

beta, betaR, Betabinom.

Examples

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