VGAM (version 1.1-12)

Zipois: Zero-Inflated Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the zero-inflated and zero-deflated 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)

Value

dzipois gives the density,

pzipois gives the distribution function,

qzipois gives the quantile function, and

rzipois generates random deviates.

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. If \(\phi\) lies in (0, 1) then this is known as the zero-inflated Poisson (ZIP) distribution. This argument may be negative to allow for 0-deflation, hence its interpretation as a probability ceases.

log

Logical. Return the logarithm of the answer?

Author

T. W. Yee

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, Gaitdpois, 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)))  # 0?
table(rzipois(100, lambda, pstr0 = pstr0))

table(qzipois(runif(100), lambda, pstr0))
round(dzipois(0:10, lambda, pstr0 = pstr0) * 100)  # Similar?

if (FALSE)  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 = paste0("ZIP(", lambda,
                      ", pstr0 = ", pstr0, ") (blue) vs",
                      " Poisson(", lambda, ") (orange)"),
        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 = paste0("ZDP(", lambda, ", pstr0 = ", newpstr0, ")",
                     " (blue) vs Poisson(", lambda, ") (orange)"),
        names.arg = as.character(x))