Generalized event count (Katz) distribution functions

Density, distribution, quantile, and random generation for the King (1989) generalized event count model with rate lambda and Katz dispersion delta (< 1 underdispersed, = 1 Poisson, > 1 overdispersed). On an unbounded support (delta >= 1) the mean is lambda and the variance-to-mean ratio is delta exactly; for delta < 1 the support is finite and the renormalized distribution's mean and variance equal those values only when lambda/(1 - delta) is an integer (the exact moments are finite sums of the pmf, as count_reg()'s siblings report them). Consistent with the estimator gec().

Usage

dgec(x, lambda, delta, max.support = 500, log = FALSE)

pgec(q, lambda, delta, max.support = 500, lower.tail = TRUE, log.p = FALSE)

qgec(p, lambda, delta, max.support = 500, lower.tail = TRUE, log.p = FALSE)

rgec(n, lambda, delta, max.support = 500)

Arguments

x, q

Vector of quantiles (non-negative integers).

lambda

Rate parameter (scalar or vector, recycled).

delta

Katz dispersion parameter (scalar).

max.support

Guard on the evaluated support.

log, log.p

Return log probabilities.

lower.tail

If TRUE (default), \(P(X \le x)\).

p

Vector of probabilities.

n

Number of draws.

Value

dgec a density, pgec a CDF, qgec a quantile, rgec a numeric vector of count draws.

See also

gec()

Examples

dgec(0:5, lambda = 3, delta = 0.7)
#> [1] 0.02824752 0.12106082 0.23347444 0.26682793 0.20012095 0.10291935
var(rgec(2000, lambda = 3, delta = 0.7)) / mean(rgec(2000, lambda = 3, delta = 0.7))
#> [1] 0.6865009