Hyper-Poisson distribution functions

Density, distribution, quantile, and random generation for the hyper-Poisson distribution of Bardwell and Crow (1964) in the mean parameterization of Saez-Castillo and Conde-Sanchez (2013): mean mu and dispersion gamma > 0, with \(P(Y = y) = \lambda^y / ((\gamma)_y\, {}_1F_1(1; \gamma; \lambda))\), \((\gamma)_y = \Gamma(\gamma + y)/\Gamma(\gamma)\), and the rate \(\lambda\) the unique solution of \(E(Y) = \mu\). gamma < 1 gives underdispersion, gamma = 1 the Poisson, gamma > 1 overdispersion. The underdispersion is bounded: as gamma -> 0 the distribution tends to 1 + Poisson(mu - 1) for mu > 1 and to a Bernoulli(mu) below, so the variance is at least mu - 1 (mu >= 1) or mu (1 - mu) (mu < 1), and Var/Mean approaches one as the mean grows (0.5 at a mean of 2, 0.9 at 10). The normalizer and the moments are exact sums over the support.

Usage

dhyperpois(x, mu, gamma, log = FALSE)

phyperpois(q, mu, gamma, lower.tail = TRUE, log.p = FALSE)

qhyperpois(p, mu, gamma, lower.tail = TRUE, log.p = FALSE)

rhyperpois(n, mu, gamma)

Arguments

x, q

Vector of quantiles (non-negative integers).

mu

Mean (scalar or vector, recycled).

gamma

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

dhyperpois a density, phyperpois a CDF, qhyperpois a quantile, rhyperpois a numeric vector of count draws.

References

Bardwell, G. E. and Crow, E. L. (1964). A two-parameter family of hyper-Poisson distributions. Journal of the American Statistical Association, 59(305), 133-141. Sáez-Castillo, A. J. and Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.

See also

count_reg()

Examples

dhyperpois(0:5, mu = 3, gamma = 0.5)
#> [1] 0.02867980 0.14422154 0.24174798 0.24313464 0.17466375 0.09759201
var(rhyperpois(2000, mu = 3, gamma = 0.5)) / 3
#> [1] 0.8686927