| NegBinomial {stats} | R Documentation |
Density, distribution function, quantile function and random
generation for the negative binomial distribution with parameters
size and prob.
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)
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. |
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]. |
The negative binomial distribution with size = n and
prob = p has density
p(x) = Gamma(x+n)/(Gamma(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.
A negative binomial distribution can arise as a mixture of Poisson
distributions with mean distributed as a
Γ (pgamma) distribution with scale parameter
(1 - prob)/prob and shape parameter size. (This
definition allows non-integer values of size.)
In this model prob = scale/(1+scale), and the mean is
size * (1 - prob)/prob.
The 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 or
n (1-p)/p^2 in the first one.
If an element of x is not integer, the result of dnbinom
is zero, with a warning.
The quantile is defined as the smallest value x such that F(x) >= p, where F is the distribution function.
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.
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.
dbinom for the binomial, dpois for the
Poisson and dgeom for the geometric distribution, which
is a special case of the negative binomial.
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, pr= 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)]))