polyaAeppli (version 2.0.2)

PolyaAeppli: Polya-Aeppli

Description

Density, distribution function, quantile function and random generation for the Polya-Aeppli distribution with parameters lambda and prob.

Usage

dPolyaAeppli(x, lambda, prob, log = FALSE)
pPolyaAeppli(q, lambda, prob, lower.tail = TRUE, log.p = FALSE)
qPolyaAeppli(p, lambda, prob, lower.tail = TRUE, log.p = FALSE)
rPolyaAeppli(n, lambda, prob)

Arguments

x

vector of quantiles

q

vector of quantiles

p

vector of probabilities

n

number of random variables to return

lambda

a vector of non-negative Poisson parameters

prob

a vector of geometric parameters between 0 and 1

log, log.p

logical; if TRUE, probabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise \(P[X > x]\)

Value

dPolyaAeppli gives the (log) density, pPolyaAepploi gives the (log) distribution function, qPolyaAeppli gives the quantile function, and rPolyaAeppli generates random deviates.

Invalid lambda or prob will terminate with an error message.

Details

A Polya-Aeppli, or geometric compound Poisson, random variable is the sum of a Poisson number of identically and independently distributed shifted geometric random variables. Its distribution (with lambda\(= \lambda\), prob\(= p\)) has density $$Prob(X = x) = e^(-\lambda)$$ for \(x = 0\); $$Prob(X = x) = e^(-\lambda) \sum_{n = 1}^y (\lambda^n)/(n!) choose(y - 1, n - 1) p^(y - n) (1 - p)^n$$ for \(x = 1, 2, \ldots\).

If an element of x is not integer, the result of dPolyaAeppli is zero, with a warning.

The quantile is right continuous: qPolyaAeppli(p, lambda, prob) is the smallest integer \(x\) such that \(P(X \le x) \ge p\).

Setting lower.tail = FALSE enables much more precise results when the default, lower.tail = TRUE would return 1, see the example below.

References

Johnson NL, Kotz S, Kemp AW (1992). Univariate Discrete Distributions. 2nd edition. Wiley, New York.

Nuel G (2008). Cumulative distribution function of a geometeric Poisson distribution. Journal of Statistical Computation and Simulation, 78(3), 385-394.

Examples

# NOT RUN {
lambda <- 8
prob <- 0.2
## Plot histogram of random sample
PAsample <- rPolyaAeppli(10000, lambda, prob)  
maxPA <- max(PAsample)
hist(PAsample, breaks=(0:(maxPA + 1)) - 0.5, freq=FALSE, 
			xlab = "x", ylab = expression(P[X](x)), main="", border="blue")
## Add plot of density function
x <- 0:maxPA
points(x, dPolyaAeppli(x, lambda, prob), type="h", lwd=2)

lambda <- 4000
prob <- 0.005
qq <- 0:10000
## Plot log of the extreme lower tail p-value
log.pp <- pPolyaAeppli(qq, lambda, prob, log.p=TRUE)
plot(qq, log.pp, type = "l", ylim=c(-lambda,0), 
  				xlab = "x", ylab = expression("log Pr(X " <= "x)"))
## Plot log of the extreme upper tail p-value
log.1minuspp <- pPolyaAeppli(qq, lambda, prob, log.p=TRUE, lower.tail=FALSE)
points(qq, log.1minuspp, type = "l", col = "red")
legend("topright", c("lower tail", "upper tail"), 
				col=c("black", "red"), lty=1, bg="white") 
# }