fipp (version 1.0.0)

dbnb: Probability density function of the BNB distribution

Description

Evaluates the probability density function of the beta-negative-binomial (BNB) distribution with a mean parameter and two shape parameters.

Usage

dbnb(x, mu, a, b, log = FALSE)

Arguments

x

vector of quantiles.

mu

mean parameter.

a

1st shape parameter.

b

2nd shape parameter.

log

logical; if TRUE, density values p are given as log(p).

Value

Numeric vector of density values.

Details

The BNB distribution has density

$$ f(x) = \frac{\Gamma(\mu + x) B(\mu + a, x + b)}{\Gamma(\mu) \Gamma(x + 1) B(a, b)}, $$ where \(\mu\) is the mean parameter and \(a\) and \(b\) are the first and second shape parameter.

References

Fr<U+00FC>hwirth-Schnatter, S., Malsiner-Walli, G., and Gr<U+00FC>n, B. (2020) Generalized mixtures of finite mixtures and telescoping sampling https://arxiv.org/abs/2005.09918

Examples

# NOT RUN {
## Similar to other d+DISTRIBUTION_NAME functions such as dnorm, it
## evaluates the density of a distribution (in this case the BNB distri)
## at point x
##
## Let's try with the density of x = 1 for BNB(1,4,3)
x <- 1
dbnb(x, mu = 1, a = 4, b = 3)

## The primary use of this function is in the closures returned from
## fipp() or nCluststers() as a prior on K-1
pmf <- nClusters(Kplus = 1:15, N = 100, type = "static",
gamma = 1, maxK = 150)

## Now evaluate above when K-1 ~ BNB(1,4,3)
pmf(priorK = dbnb, priorKparams = list(mu = 1, a = 4, b = 3))

## Compare the result with the case when K-1 ~ Pois(1)
pmf(priorK = dpois, priorKparams = list(lambda = 1))

## Although both BNB(1,4,3) and Pois(1) have 1 as their mean, the former
## has a fatter rhs tail. We see that it is reflected in the induced prior 
## on K+ as well

# }