BetaBinom: Beta-binomial distribution

In extraDistr: Additional Univariate and Multivariate Distributions

Description

Probability mass function and random generation for the beta-binomial distribution.

Usage

dbbinom(x, size, alpha = 1, beta = 1, log = FALSE)

pbbinom(q, size, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rbbinom(n, size, alpha = 1, beta = 1)

Arguments

x, q	vector of quantiles.
size	number of trials (zero or more).
alpha, beta	non-negative parameters of the beta distribution.
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].
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If p ~ Beta(α, β) and X ~ Binomial(n, p), then X ~ BetaBinomial(n, α, β).

Probability mass function

f(x) = choose(n, x) * B(x+α, n-x+β) / B(α, β)

Cumulative distribution function is calculated using recursive algorithm that employs the fact that Γ(x) = (x - 1)!, and B(x, y) = (Γ(x)Γ(y))/Γ(x+y) , and that choose(n,k) = prod((n+1-(1:k))/(1:k)) . This enables re-writing probability mass function as

f(x) = prod((n+1-(1:x))/(1:x)) * (((α+x-1)!*(β+n-x-1)!)/((α+β+n+1)!)) / B(α, β)

what makes recursive updating from x to x+1 easy using the properties of factorials

f(x+1) = prod((n+1-(1:x))/(1:x)) * ((n+1-x+1)/(x+1)) * (((α+x-1)!*(α+x)*(β+n-x-1)!/(β+n-x))/((α+β+n+1)!*(α+β+n))) / B(α, β)

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x) = f(0)+...+f(x)

See Also

Beta, Binomial

Examples

x <- rbbinom(1e5, 1000, 5, 13)
xx <- 0:1000
hist(x, 100, freq = FALSE)
lines(xx-0.5, dbbinom(xx, 1000, 5, 13), col = "red")
hist(pbbinom(x, 1000, 5, 13))
xx <- seq(0, 1000, by = 0.1)
plot(ecdf(x))
lines(xx, pbbinom(xx, 1000, 5, 13), col = "red", lwd = 2)

extraDistr documentation built on Sept. 7, 2020, 5:09 p.m.