Pospois {VGAM}R Documentation

Positive-Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the positive-Poisson distribution.

Usage

dpospois(x, lambda)
ppospois(q, lambda)
qpospois(p, lambda)
rpospois(n, lambda)

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 (of an ordinary Poisson distribution). Short vectors are recycled.

Details

The positive-Poisson distribution is a Poisson distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is

lambda / (1-exp(-lambda)).

As lambda increases, the positive-Poisson and Poisson distributions become more similar. Unlike similar functions for the Poisson distribution, a zero value of lambda is not permitted here.

Value

dpospois gives the density, ppospois gives the distribution function, qpospois gives the quantile function, and rpospois generates random deviates.

Note

For rpospois, the arguments of the function are fed into rpois until n positive values are obtained. This may take a long time if lambda has values close to 0.

The family function pospoisson estimates lambda by maximum likelihood estimation.

Author(s)

T. W. Yee

See Also

pospoisson, zapoisson, rpois.

Examples

lambda = 2
y = rpospois(n=1000, lambda)
table(y)
mean(y)  # Sample mean
lambda / (1-exp(-lambda)) # Population mean

(i = dpospois(0:7, lambda))
cumsum(i) - ppospois(0:7, lambda)  # Should be 0s
table(rpospois(100, lambda))

table(qpospois(runif(1000), lambda))
round(dpospois(1:10, lambda) * 1000) # Should be similar

## Not run: 
x = 0:7
barplot(rbind(dpospois(x, lambda), dpois(x, lambda)),
        beside = TRUE, col = c("blue","green"),
        main=paste("Positive Poisson(", lambda, ") (blue) vs",
        " Poisson(", lambda, ") (green)", sep=""),
        names.arg = as.character(x), las=1, lwd=2)
## End(Not run)

[Package VGAM version 0.7-7 Index]