VGAM (version 0.8-7)

Zipois: Zero-Inflated Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the zero-inflated Poisson distribution with parameter pstr0.

Usage

dzipois(x, lambda, pstr0 = 0, log = FALSE)
pzipois(q, lambda, pstr0 = 0)
qzipois(p, lambda, pstr0 = 0)
rzipois(n, lambda, pstr0 = 0)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Must be a single positive integer.

lambda

Vector of positive means.

pstr0

Probability of a structural zero (i.e., ignoring the Poisson distribution), called $\phi$. The default value of $\phi = 0$ corresponds to the response having an ordinary Poisson distribution.

log

Logical. Return the logarithm of the answer?

Value

• dzipois gives the density, pzipois gives the distribution function, qzipois gives the quantile function, and rzipois generates random deviates.

Details

The probability function of $Y$ is 0 with probability $\phi$, and $Poisson(\lambda)$ with probability $1-\phi$. Thus $$P(Y=0) =\phi + (1-\phi) P(W=0)$$ where $W$ is distributed $Poisson(\lambda)$.

See Also

zipoisson, dpois, rzinegbin.

Examples

lambda = 3; pstr0 = 0.2; x = (-1):7
(ii = dzipois(x, lambda, pstr0 = pstr0))
max(abs(cumsum(ii) - pzipois(x, lambda, pstr0 = pstr0))) # Should be 0
table(rzipois(100, lambda, pstr0 = pstr0))

table(qzipois(runif(100), lambda, pstr0))
round(dzipois(0:10, lambda, pstr0 = pstr0) * 100) # Should be similar

x = 0:10
par(mfrow = c(2, 1)) # Zero-inflated Poisson
barplot(rbind(dzipois(x, lambda, pstr0 = pstr0), dpois(x, lambda)),
        beside = TRUE, col = c("blue","orange"),
        main = paste("ZIP(", lambda, ", pstr0 = ", pstr0, ") (blue) vs",
                     " Poisson(", lambda, ") (orange)", sep = ""),
        names.arg = as.character(x))

deflat_limit = -1 / expm1(lambda) # Zero-deflated Poisson
newpstr0 = round(deflat_limit / 1.5, 3)
barplot(rbind(dzipois(x, lambda, pstr0 = newpstr0),
                dpois(x, lambda)),
        beside = TRUE, col = c("blue","orange"),
        main = paste("ZDP(", lambda, ", pstr0 = ", newpstr0, ") (blue) vs",
                     " Poisson(", lambda, ") (orange)", sep = ""),
        names.arg = as.character(x))