VGAM (version 1.1-9)

pospoisson: Positive Poisson Distribution Family Function

Description

Fits a positive Poisson distribution.

Usage

pospoisson(link = "loglink", type.fitted = c("mean", "lambda",
           "prob0"), expected = TRUE, ilambda = NULL, imethod = 1,
           zero = NULL, gt.1 = FALSE)

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm,

rrvglm and vgam.

Arguments

link

Link function for the usual mean (lambda) parameter of an ordinary Poisson distribution. See Links for more choices.

expected

Logical. Fisher scoring is used if expected = TRUE, else Newton-Raphson.

ilambda, imethod, zero

See CommonVGAMffArguments for information.

type.fitted

See CommonVGAMffArguments for details.

gt.1

Logical. Enforce lambda > 1? The default is to enforce lambda > 0.

Warning

Under- or over-flow may occur if the data is ill-conditioned.

Author

Thomas W. Yee

Details

The positive Poisson distribution is the ordinary Poisson distribution but with the probability of zero being zero. Thus the other probabilities are scaled up (i.e., divided by \(1-P[Y=0]\)). The mean, \(\lambda / (1 - \exp(-\lambda))\), can be obtained by the extractor function fitted applied to the object.

A related distribution is the zero-inflated Poisson, in which the probability \(P[Y=0]\) involves another parameter \(\phi\). See zipoisson.

References

Coleman, J. S. and James, J. (1961). The equilibrium size distribution of freely-forming groups. Sociometry, 24, 36--45.

See Also

Gaitdpois, gaitdpoisson, posnegbinomial, poissonff, zapoisson, zipoisson, simulate.vlm, otpospoisson, Pospois.

Examples

# Data from Coleman and James (1961)
cjdata <- data.frame(y = 1:6, freq = c(1486, 694, 195, 37, 10, 1))
fit <- vglm(y ~ 1, pospoisson, data = cjdata, weights = freq)
Coef(fit)
summary(fit)
fitted(fit)

pdata <- data.frame(x2 = runif(nn <- 1000))  # Artificial data
pdata <- transform(pdata, lambda = exp(1 - 2 * x2))
pdata <- transform(pdata, y1 = rgaitdpois(nn, lambda, truncate = 0))
with(pdata, table(y1))
fit <- vglm(y1 ~ x2, pospoisson, data = pdata, trace = TRUE, crit = "coef")
coef(fit, matrix = TRUE)