BetaNegBinom: Beta-negative binomial distribution

In extraDistr: Additional Univariate and Multivariate Distributions

Description

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

Usage

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

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

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

Arguments

x, q	vector of quantiles.
size	number of trials (zero or more). Must be strictly positive, need not be integer.
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 ~ NegBinomial(r, p), then X ~ BetaNegBinomial(r, α, β).

Probability mass function

f(x) = Γ(r+x)/(x! Γ(r)) * B(α+r, β+x) / 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) . This enables re-writing probability mass function as

f(x) = ((r+x-1)!)/(x!*Γ(r))*(((α+r-1)!*(β+x-1)!)/((α+β+r+x-1)!))/B(α,β)

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

f(x+1) = ((r+x-1)!*(r+x))/(x!*(x+1)*Γ(r))*(((α+r-1)!*(β+x-1)!*(β+x))/((α+β+r+x-1)!*(α+β+r+x)))/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, NegBinomial

Examples

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

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