VGAM (version 1.0-1)

Posnegbin: Positive-Negative Binomial Distribution

Description

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

Usage

dposnegbin(x, size, prob = NULL, munb = NULL, log = FALSE)
pposnegbin(q, size, prob = NULL, munb = NULL)
qposnegbin(p, size, prob = NULL, munb = NULL)
rposnegbin(n, size, prob = NULL, munb = NULL)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Fed into runif.

size, prob, munb, log

Same arguments as that of an ordinary negative binomial distribution (see dnbinom). Some arguments have been renamed slightly. Short vectors are recycled. The parameter 1/siz

Value

• dposnegbin gives the density, pposnegbin gives the distribution function, qposnegbin gives the quantile function, and rposnegbin generates $n$ random deviates.

Details

The positive-negative binomial distribution is a negative binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$\mu / (1-p(0))$$ where $\mu$ the mean of an ordinary negative binomial distribution.

References

Welsh, A. H., Cunningham, R. B., Donnelly, C. F. and Lindenmayer, D. B. (1996) Modelling the abundances of rare species: statistical models for counts with extra zeros. Ecological Modelling, 88, 297--308.

See Also

posnegbinomial, zanegbinomial, zinegbinomial, rnbinom.

Examples

munb <- 5; size <- 4; n <- 1000
table(y <- rposnegbin(n, munb = munb, size = size))
mean(y)  # sample mean
munb / (1 - (size / (size + munb))^size)  # population mean
munb / pnbinom(0, mu = munb, size = size, lower.tail = FALSE)  # same as before

x <- (-1):17
(ii <- dposnegbin(x, munb = munb, size = size))
max(abs(cumsum(ii) - pposnegbin(x, munb = munb, size = size)))  # Should be 0

x <- 0:10
barplot(rbind(dposnegbin(x, munb = munb, size = size),
                 dnbinom(x, mu   = munb, size = size)),
        beside = TRUE, col = c("blue","green"),
        main = paste("dposnegbin(munb = ", munb, ", size = ", size, ") (blue) vs",
                     " dnbinom(mu = ", munb, ", size = ", size, ") (green)", sep = ""),
        names.arg = as.character(x))

# Another test for pposnegbin()
nn <- 5000
mytab <- cumsum(table(rposnegbin(nn, munb = munb, size = size))) / nn
myans <- pposnegbin(sort(as.numeric(names(mytab))), munb = munb, size = size)
max(abs(mytab - myans))  # Should be 0