PolyaAeppli {polyaAeppli} | R Documentation |
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 |
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.
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.
Author(s)
Conrad Burden
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
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")