Binomialintersection: The Binomial Intersection Distribution

Description

Density, distribution function, quantile function and random generation for the binomial intersection distribution.

Usage

dbint(n, A, range = NULL, log = FALSE)
pbint(n, A, vals, upper.tail = TRUE, log.p = FALSE)
qbint(p, n, A, upper.tail = TRUE, log.p = FALSE)
rbint(num = 5, n, A)

Arguments

An integer specifying the number of categories in the urns.

A vector of integers specifying the numbers of balls drawn from each urn. The length of the vector equals the number of urns.

A probability between 0 and 1.

num

An integer specifying the number of random numbers to generate. Defaults to 5.

range, vals

A vector of integers specifying the intersection sizes for which probabilities (dhint) or cumulative probabilites (phint) should be computed (can be a single number). If range is NULL (default) then probabilities will be returned over the entire range of possible values.

log, log.p

Logical. If TRUE, probabilities p are given as log(p). Defaults to FALSE.

upper.tail

Logical. If TRUE, probabilities are P(X >= v), else P(X

Value

dbint, pbint, and qbint return a data frame with two columns: v, the intersection size, and p, the associated p-values. rbint returns an integer vector of random samples based on the binomial intersection distribution.

Details

The binomial intersection distribution is given by $$ P(X = v|N) = {b \choose v} \left(\prod_{i=1}^{N-1} p_{i}\right)^{v} \left(1 - \prod_{i=1}^{N-1} p_{i}\right)^{b-v} $$ where b gives the sample size which is smallest. This is an approximation for the hypergeometric intersection distribution when $n$ is large and $b$ is small relative to the samples taken from the $N-1$ other urns.

References

Kalinka, A.T. (2013). The probability of drawing intersections: extending the hypergeometric distribution. arXiv.1305.0717

Examples

Run this code

## Generate the distribution of intersections sizes:
dd <- dbint(20, c(10, 12, 11, 14))
## Restrict the range of intersections.
dd <- dbint(20, c(10, 12), range = 0:5)
## Generate cumulative probabilities.
pp <- pbint(29, c(15, 8), vals = 5)
pp <- pbint(29, c(15, 8), vals = 2, upper.tail = FALSE)
## Extract quantiles:
qq <- qbint(0.15, 23, c(12, 10))
## Generate random samples from Binomial intersection distributions.
rr <- rbint(num = 10, 18, c(9, 14))

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