Gamma-count distribution functions

Density, distribution, quantile, and random generation for Winkelmann's (1995) gamma-count distribution: the number of events in a unit interval when the waiting times between events are independent Gamma variables with shape alpha and rate alpha * mu, so that mu is the long-run event rate. alpha > 1 (waiting times more regular than exponential) gives underdispersion, alpha = 1 is the Poisson, alpha < 1 overdispersion; the variance-to-mean ratio is approximately 1/alpha. The exact mean and variance are finite sums of incomplete-gamma terms and are what count_reg(family = "gammacount") reports through predict() and fitted().

Usage

dgammacount(x, mu, alpha, log = FALSE)

pgammacount(q, mu, alpha, lower.tail = TRUE, log.p = FALSE)

qgammacount(p, mu, alpha, lower.tail = TRUE, log.p = FALSE)

rgammacount(n, mu, alpha)

Arguments

x, q

Vector of quantiles (non-negative integers).

mu

Rate parameter (scalar or vector, recycled).

alpha

Dispersion parameter (scalar, positive).

log, log.p

Return log probabilities.

lower.tail

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

p

Vector of probabilities.

n

Number of draws.

Value

dgammacount a density, pgammacount a CDF, qgammacount a quantile, rgammacount a numeric vector of count draws.

References

Winkelmann, R. (1995). Duration dependence and dispersion in count-data models. Journal of Business & Economic Statistics, 13(4), 467-474.

See also

count_reg()

Examples

dgammacount(0:5, mu = 3, alpha = 2)
#> [1] 0.01735127 0.13385262 0.29447576 0.29830012 0.17209622 0.06383205
var(rgammacount(2000, mu = 3, alpha = 2)) / 3
#> [1] 0.5035768