R: The Negative Binomial Distribution

NegBinomial {stats} | R Documentation

## The Negative Binomial Distribution

### Description

Density, distribution function, quantile function and random generation for the negative binomial distribution with parameters `size` and `prob`.

### Usage

    dnbinom(x, size, prob, mu, log = FALSE)
    pnbinom(q, size, prob, mu, lower.tail = TRUE, log.p = FALSE)
    qnbinom(p, size, prob, mu, lower.tail = TRUE, log.p = FALSE)
    rnbinom(n, size, prob, mu)

### Arguments

`x` | vector of (non-negative integer) quantiles.
`q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`size` | target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer.
`prob` | probability of success in each trial. `0 < prob <= 1`.
`mu` | alternative parametrization via mean: see ‘Details’.
`log, log.p` | logical; if TRUE, probabilities p are given as log(p).
`lower.tail` | logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Details

The negative binomial distribution with `size` = n and `prob` = p has density

Γ(x+n)/(Γ(n) x!) p^n (1-p)^x

for x = 0, 1, 2, …, n > 0 and 0 < p ≤ 1.

This represents the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached. The mean is n(1-p)/p and variance n(1-p)/p^2.

A negative binomial distribution can also arise as a mixture of Poisson distributions with mean distributed as a gamma distribution (see `pgamma`) with scale parameter `(1 - prob)/prob` and shape parameter `size`. (This definition allows non-integer values of `size`.)

An alternative parametrization (often used in ecology) is by the mean `mu`, and `size`, the dispersion parameter, where `prob` = `size/(size+mu)`. The variance is `mu + mu^2/size` in this parametrization.

If an element of `x` is not integer, the result of `dnbinom` is zero, with a warning.

The case `size == 0` is the distribution concentrated at zero. This is the limiting distribution for `size` approaching zero, even if `mu` rather than `prob` is held constant. Notice though, that the mean of the limit distribution is 0, whatever the value of `mu`.

The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.

### Value

`dnbinom` gives the density, `pnbinom` gives the distribution function, `qnbinom` gives the quantile function, and `rnbinom` generates random deviates.

Invalid `size` or `prob` will result in return value `NaN`, with a warning.

The length of the result is determined by `n` for `rnbinom`, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used.

### Source

`dnbinom` computes via binomial probabilities, using code contributed by Catherine Loader (see `dbinom`).

`pnbinom` uses `pbeta`.

`qnbinom` uses the Cornish–Fisher Expansion to include a skewness correction to a normal approximation, followed by a search.

`rnbinom` uses the derivation as a gamma mixture of Poissons, see

Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer-Verlag, New York. Page 480.

### See Also

Distributions for standard distributions, including `dbinom` for the binomial, `dpois` for the Poisson and `dgeom` for the geometric distribution, which is a special case of the negative binomial.

### Examples

    require(graphics)
    x <- 0:11
    dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1
    126 /  dnbinom(0:8, size  = 2, prob  = 1/2) #- theoretically integer

    ## Cumulative ('p') = Sum of discrete prob.s ('d');  Relative error :
    summary(1 - cumsum(dnbinom(x, size = 2, prob = 1/2)) /
                      pnbinom(x, size  = 2, prob = 1/2))

    x <- 0:15
    size <- (1:20)/4
    persp(x, size, dnb <- outer(x, size, function(x,s) dnbinom(x, s, prob = 0.4)),
          xlab = "x", ylab = "s", zlab = "density", theta = 150)
    title(tit <- "negative binomial density(x,s, pr = 0.4)  vs.  x & s")

    image  (x, size, log10(dnb), main = paste("log [", tit, "]"))
    contour(x, size, log10(dnb), add = TRUE)

    ## Alternative parametrization
    x1 <- rnbinom(500, mu = 4, size = 1)
    x2 <- rnbinom(500, mu = 4, size = 10)
    x3 <- rnbinom(500, mu = 4, size = 100)
    h1 <- hist(x1, breaks = 20, plot = FALSE)
    h2 <- hist(x2, breaks = h1$breaks, plot = FALSE)
    h3 <- hist(x3, breaks = h1$breaks, plot = FALSE)
    barplot(rbind(h1$counts, h2$counts, h3$counts),
            beside = TRUE, col = c("red","blue","cyan"),
            names.arg = round(h1$breaks[-length(h1$breaks)]))

[Package stats version 3.1.2 Index]