R: Positive-Negative Binomial Distribution

Posnegbin {VGAM} | R Documentation

## 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,
               lower.tail = TRUE, log.p = FALSE)
    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/size` is known as a dispersion parameter; as `size` approaches infinity, the negative binomial distribution approaches a Poisson distribution. Note that `prob` must lie in (0,1), otherwise a `NaN` is returned.
`log.p, lower.tail` | Same arguments as that of an ordinary negative binomial distribution (see `pnbinom`).

### 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

munb / (1-p(0))

where munb the mean of an ordinary negative binomial distribution.

### Value

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

### Author(s)

T. W. Yee

### 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

    ## Not run:
    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))
    ## End(Not run)

    # 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

[Package VGAM version 1.0-3 Index]