VGAM (version 1.1-1)

Posbinom: Positive-Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the positive-binomial distribution.

Usage

dposbinom(x, size, prob, log = FALSE)
pposbinom(q, size, prob)
qposbinom(p, size, prob)
rposbinom(n, size, prob)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Fed into runif.

size

number of trials. It is the \(N\) symbol in the formula given in posbinomial and should be positive.

prob

probability of success on each trial. Should be in \((0,1)\).

log

See dbinom.

Value

dposbinom gives the density, pposbinom gives the distribution function, qposbinom gives the quantile function, and rposbinom generates random deviates.

Details

The positive-binomial distribution is a binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$\mu / (1-(1-\mu)^N)$$ where \(\mu\) is the argument prob above. As \(\mu\) increases, the positive-binomial and binomial distributions become more similar. Unlike similar functions for the binomial distribution, a zero value of prob is not permitted here.

See Also

posbinomial, dposbern, dgentbinom, zabinomial, zibinomial, rbinom.

Examples

# NOT RUN {
prob <- 0.2; size <- 10
table(y <- rposbinom(n = 1000, size, prob))
mean(y)  # Sample mean
size * prob / (1 - (1 - prob)^size)  # Population mean

(ii <- dposbinom(0:size, size, prob))
cumsum(ii) - pposbinom(0:size, size, prob)  # Should be 0s
table(rposbinom(100, size, prob))

table(qposbinom(runif(1000), size, prob))
round(dposbinom(1:10, size, prob) * 1000)  # Should be similar

# }
# NOT RUN {
 barplot(rbind(dposbinom(x = 0:size, size, prob),
                           dbinom(x = 0:size, size, prob)),
        beside = TRUE, col = c("blue", "green"),
        main = paste("Positive-binomial(", size, ",",
                      prob, ") (blue) vs",
        " Binomial(", size, ",", prob, ") (green)", sep = ""),
        names.arg = as.character(0:size), las = 1) 
# }
# NOT RUN {
# Simulated data example
nn <- 1000; sizeval1 <- 10; sizeval2 <- 20
pdata <- data.frame(x2 = seq(0, 1, length = nn))
pdata <- transform(pdata, prob1  = logitlink(-2 + 2 * x2, inverse = TRUE),
                          prob2  = logitlink(-1 + 1 * x2, inverse = TRUE),
                          sizev1 = rep(sizeval1, len = nn),
                          sizev2 = rep(sizeval2, len = nn))
pdata <- transform(pdata, y1 = rposbinom(nn, size = sizev1, prob = prob1),
                          y2 = rposbinom(nn, size = sizev2, prob = prob2))
with(pdata, table(y1))
with(pdata, table(y2))
# Multiple responses
fit2 <- vglm(cbind(y1, y2) ~ x2, posbinomial(multiple.responses = TRUE),
             trace  = TRUE, data = pdata, weight = cbind(sizev1, sizev2))
coef(fit2, matrix = TRUE)
# }