ZIPoisson: Create a zero-inflated Poisson distribution

In distributions3: Probability Distributions as S3 Objects

Create a zero-inflated Poisson distribution

Description

Zero-inflated Poisson distributions are frequently used to model counts with many zero observations.

Usage

ZIPoisson(lambda, pi)

Arguments

lambda	Parameter of the Poisson component of the distribution. Can be any positive number.
pi	Zero-inflation probability, can be any value in ⁠[0, 1]⁠.

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-inflated Poisson random variable with parameter lambda = \lambda.

Support: \{0, 1, 2, 3, ...\}

Mean: (1 - \pi) \cdot \lambda

Variance: (1 - \pi) \cdot \lambda \cdot (1 + \pi \cdot \lambda)

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

P(X = k) = \pi \cdot I_{0}(k) + (1 - \pi) \cdot f(k; \lambda)

where I_{0}(k) is the indicator function for zero and f(k; \lambda) is the p.m.f. of the Poisson distribution.

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

P(X \le k) = \pi + (1 - \pi) \cdot F(k; \lambda)

where F(k; \lambda) is the c.d.f. of the Poisson distribution.

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

E(e^{tX}) = \pi + (1 - \pi) \cdot e^{\lambda (e^t - 1)}

Value

A ZIPoisson object.

See Also

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

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 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)

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.