# Polya-Aeppli Density, distribution function, quantile function and random generation for the Polya-Aeppli distribution with parameters lambda and prob. ```r 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 $P[X > x]$ ## 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. ## Returns `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. ## 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. ## Author(s) Conrad Burden ## Examples ```r 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") ```

dPolyaAeppli function