Double Poisson distribution functions

Density, distribution, quantile, and random generation for Efron's (1986) double Poisson distribution with location mu and dispersion theta, normalized by the exact sum rather than Efron's closed-form approximation. theta > 1 gives underdispersion, theta = 1 is the Poisson, theta < 1 overdispersion; the variance-to-mean ratio is approximately 1/theta, and the mean is close to (but not exactly) mu. The exact mean and variance are what count_reg(family = "doublepois") reports through predict() and fitted().

Usage

ddoublepois(x, mu, theta, log = FALSE)

pdoublepois(q, mu, theta, lower.tail = TRUE, log.p = FALSE)

qdoublepois(p, mu, theta, lower.tail = TRUE, log.p = FALSE)

rdoublepois(n, mu, theta)

Arguments

x, q

Vector of quantiles (non-negative integers).

mu

Location parameter (scalar or vector, recycled).

theta

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

ddoublepois a density, pdoublepois a CDF, qdoublepois a quantile, rdoublepois a numeric vector of count draws.

References

Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81(395), 709-721.

See also

count_reg()

Examples

ddoublepois(0:5, mu = 3, theta = 2)
#> [1] 0.003568911 0.087311745 0.267005171 0.322575691 0.208081004 0.083404477
sum(ddoublepois(0:60, mu = 3, theta = 2))
#> [1] 1