Create a zero-truncated negative binomial distribution

Zero-truncated negative binomial distributions are frequently used to model counts where zero observations cannot occur or have been excluded.

Usage

ZTNegativeBinomial(mu, theta)

Arguments

mu

Location parameter of the negative binomial component of the distribution. Can be any positive number.

theta

Overdispersion parameter of the negative binomial component of the distribution. Can be any positive number.

Value

A ZTNegativeBinomial object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let \(X\) be a zero-truncated negative binomial random variable with parameter mu = \(\mu\).

Support: \(\{1, 2, 3, ...\}\)

Mean: $$ \mu \cdot \frac{1}{1 - F(0; \mu, \theta)} $$

where \(F(k; \mu, \theta)\) is the c.d.f. of the NegativeBinomial distribution.

Variance: \(m \cdot (\mu + 1 - m)\), where \(m\) is the mean above.

Probability mass function (p.m.f.):

$$ P(X = k) = \frac{f(k; \mu, \theta)}{1 - F(0; \mu, \theta)} $$

where \(f(k; \mu, \theta)\) is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.):

$$ P(X = k) = \frac{F(k; \mu, \theta)}{1 - F(0; \mu, \theta)} $$

Moment generating function (m.g.f.):

Omitted for now.

See also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), Poisson(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTPoisson()

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X
#> [1] "ZTNegativeBinomial(mu = 2.5, theta = 1)"

## standard functions
pdf(X, 0:8)
#> [1] 0.00000000 0.28571429 0.20408163 0.14577259 0.10412328 0.07437377 0.05312412
#> [8] 0.03794580 0.02710414
cdf(X, 0:8)
#> [1] 0.0000000 0.2857143 0.4897959 0.6355685 0.7396918 0.8140656 0.8671897
#> [8] 0.9051355 0.9322396
quantile(X, seq(0, 1, by = 0.25))
#> [1]   1   1   3   5 Inf

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))
#> [1] 3

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)


## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)