The Generalized Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the generalized binomial distribution with parameter vectors size and prob.

Usage

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dgbinom(x, size, prob, log = FALSE)
pgbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
qgbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
rgbinom(N, size, prob)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

N

number of observations.

size

vector of the number of trials for each type.

prob

vector of the success probabilities for each type.

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 generalized binomial distribution with size=c(n1,… ,nr) and prob=c(p1,...,pr) is the sum of r binomially distributed random variables with different pi (and, in case, with different ni):

Z=∑ Zi, Z ~ gbinom(size,prob), with Zi ~ binom(ni,pi), i=1,...,r.

Since the sum of Bernoulli distributed random variables is binomially distributed, Z can be also defined as:

Z=∑ ∑ Zij, with Zij ~ binom(1,pi), j=1,...,ni.

The pmf is obtained by an algorithm which is based on the convolution of Bernoulli distributions. See the references below for further information.

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

rgbinom uses the inversion method (see Devroye, 1986).

Value

dgbinom gives the pmf, pgbinom gives the cdf, qgbinom gives the quantile function and rgbinom generates random deviates.

Note

If size contains just one trial number and prob one success probability, then the generalized binomial distribution results in the binomial distribution.

The generalized binomial distribution described here is also known as Poisson-binomial distribution. See the link below to the package poibin for further information.

References

D.Kurz, H.Lewitschnig, J.Pilz, Decision-Theoretical Model for Failures which are Tackled by Countermeasures, IEEE Transactions on Reliability, Vol. 63, No. 2, June 2014.

K.J. Klauer, Kriteriumsorientierte Tests, Verlag fuer Psychologie, Hogrefe, 1987, Goettingen, p. 208 ff.

M.Fisz, Wahrscheinlichkeitsrechnung und mathematische Statistik, VEB Deutscher Verlag der Wissenschaften, 1973, p. 164 ff.

L.Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, 1986, p. 85 ff.

See Also

ppoibin, for another implementation of this distribution.
dbinom

Examples

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## n=10 defect devices, divided in 3 failure types n1=2, n2=5, n3=3.
## 3 countermeasures with effectivities p1=0.8, p2=0.7, p3=0.3 are available.
## use dgbinom() to get the probabilities for x=0,...,10 failures solved.
dgbinom(x=c(0:10),size=c(2,5,3),prob=c(0.8,0.7,0.3))

## generation of N=100000 random values
rgbinom(100000,size=c(2,5,3),prob=c(0.8,0.7,0.3))


## n1=100, n2=100, n3=200, p1=0.001, p2=0.005, p3=0.01 
dgbinom(c(0:2),size=c(100,100,200),prob=c(0.001,0.005,0.01))
#  0.07343377 0.19260317 0.25173556
pgbinom(2,size=c(100,100,200),prob=c(0.001,0.005,0.01),lower.tail=FALSE)
# 0.4822275