HurdlePoisson: Create a hurdle Poisson distribution

In distributions3: Probability Distributions as S3 Objects

Create a hurdle Poisson distribution

Description

Hurdle Poisson distributions are frequently used to model counts with many zero observations.

Usage

HurdlePoisson(lambda, pi)

Arguments

lambda	Parameter of the Poisson component of the distribution. Can be any positive number.
pi	Zero-hurdle 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 hurdle Poisson random variable with parameter lambda = \lambda.

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

Mean:

\lambda \cdot \frac{\pi}{1 - e^{-\lambda}}

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

Probability mass function (p.m.f.): P(X = 0) = 1 - \pi and for k > 0

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

where f(k; \lambda) is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.): P(X \le 0) = 1 - \pi and for k > 0

P(X \le k) = 1 - \pi + \pi \cdot \frac{F(k; \lambda) - F(0; \lambda)}{1 - F(0; \lambda)}

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

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

Omitted for now.

Value

A HurdlePoisson object.

See Also

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

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
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.