oipospoisUC: One-Inflated Positive Poisson Distribution

In VGAMdata: Data Supporting the 'VGAM' Package

One-Inflated Positive Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the one-inflated positive Poisson distribution with parameter pstr1.

Usage

doipospois(x, lambda, pstr1 = 0, log = FALSE)
poipospois(q, lambda, pstr1 = 0)
qoipospois(p, lambda, pstr1 = 0)
roipospois(n, lambda, pstr1 = 0)

Arguments

x, p, q, n	Same as Pospois.
lambda	Vector of positive means.
pstr1	Probability of a structural one (i.e., ignoring the positive Poisson distribution), called \phi. The default value of \phi = 0 corresponds to the response having a positive Poisson distribution.
log	Logical. Return the logarithm of the answer?

Details

The probability function of Y is 1 with probability \phi, and PosPoisson(\lambda) with probability 1-\phi. Thus

P(Y=1) =\phi + (1-\phi) P(W=1)

where W is distributed as a positive Poisson(\lambda) random variate.

Value

doipospois gives the density, poipospois gives the distribution function, qoipospois gives the quantile function, and roipospois generates random deviates.

Note

The argument pstr1 is recycled to the required length, and usually has values which lie in the interval [0,1].

These functions actually allow for the zero-deflated Poisson distribution. Here, pstr1 is also permitted to lie in the interval [-lambda / (expm1(lambda) - lambda), 0]. The resulting probability of a unit count is less than the nominal positive Poisson value, and the use of pstr1 to stand for the probability of a structural 1 loses its meaning. When pstr1 equals -lambda / (expm1(lambda) - lambda) this corresponds to the 0- and 1-truncated Poisson distribution.

Author(s)

T. W. Yee

See Also

Pospois, oapospoisson, oipospoisson, otpospoisson, pospoisson, dpois, poissonff.

Examples

lambda <- 3; pstr1 <- 0.2; x <- (-1):7
(ii <- doipospois(x, lambda, pstr1 = pstr1))
table(roipospois(100, lambda, pstr1 = pstr1))
round(doipospois(1:10, lambda, pstr1 = pstr1) * 100)  # Similar?

## Not run:  x <- 0:10
par(mfrow = c(2, 1))  # One-Inflated Positive Poisson
barplot(rbind(doipospois(x, lambda, pstr1 = pstr1),
                dpospois(x, lambda)),
   beside = TRUE, col = c("blue", "orange"),
   main = paste0("OIPP(", lambda, ", pstr1 = ", pstr1,
                 ") (blue) vs PosPoisson(", lambda, ") (orange)"),
   names.arg = as.character(x))
# 0-deflated Pos Poisson:
deflat.limit <- -lambda / (expm1(lambda) - lambda)
newpstr1 <- round(deflat.limit, 3) + 0.001  # Near the boundary
barplot(rbind(doipospois(x, lambda, pstr1 = newpstr1),
                dpospois(x, lambda)),
   beside = TRUE, col = c("blue","orange"),
   main = paste0("ODPP(", lambda, ", pstr1 = ", newpstr1,
                 ") (blue) vs PosPoisson(", lambda, ") (orange)"),
   names.arg = as.character(x)) 
## End(Not run)

VGAMdata documentation built on Sept. 18, 2023, 9:08 a.m.