Generalized Poisson Version 2 Distribution

These functions provide the density function, distribution function, quantile function, and random number generation for the Generalized Poisson Version 2 (GP-2) Distribution.

Usage

dgp2(x, mu = 1, alpha = 0, log = FALSE)

pgp2(q, mu = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

qgp2(p, mu = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

rgp2(n, mu = 1, alpha = 0)

Arguments

x

numeric value or a vector of values.

mu

numeric value or vector of mean values for the distribution (the values have to be greater than 0).

alpha

numeric value or vector of values for the dispersion parameter of the distribution. Values may be negative, zero, or positive, provided that \(1 + \alpha \mu > 0\).

log

logical; if TRUE, probabilities are given as log(p).

q

quantile or a vector of quantiles.

lower.tail

logical; if TRUE, probabilities p are \(P[X\leq x]\) otherwise, \(P[X>x]\).

log.p

logical; if TRUE, probabilities p are given as log(p).

p

probability or a vector of probabilities.

n

the number of random numbers to generate.

Value

dgp2 gives the density, pgp2 gives the distribution function, qgp2 gives the quantile function, and rgp2 generates random deviates.

The length of the result is determined by n for rgp2, and is the maximum of the lengths of the numerical arguments for the other functions.

Details

dgp2 computes the density (PMF) of the Generalized Poisson Version 2 Distribution.

pgp2 computes the CDF of the Generalized Poisson Version 2 Distribution.

qgp2 computes the quantile function of the Generalized Poisson Version 2 Distribution.

rgp2 generates random numbers from the Generalized Poisson Version 2 Distribution.

The probability mass function (PMF) for the Generalized Poisson Version 2 distribution (GP-2) is: $$ f(y|\mu,\alpha)= \frac{\mu(\mu+\alpha\mu y)^{y-1} \exp\left(-\frac{\mu+\alpha\mu y}{1+\alpha\mu}\right)} {(1+\alpha\mu)^y y!} $$

where \(\mu>0\) is the mean and \(y\) is a non-negative integer. This formulation uses the mean directly.

The variance of the GP-2 distribution is: $$\sigma^2=\mu(1+\alpha\mu)^2$$

If \(\alpha > 0\), the distribution is overdispersed. If \(\alpha = 0\), the distribution is equidispersed and reduces to the ordinary Poisson distribution. If \(\alpha < 0\) and \(1 + \alpha\mu > 0\), the distribution is underdispersed and has finite support.

Under the underdispersed case, the largest possible value of \(y\) is bounded above by \(\left\lceil -1/\alpha \right\rceil - 1\), i.e., the largest integer satisfying \(1 + \alpha y > 0\).

References

Consul, PoC, and Felix Famoye. "Generalized Poisson regression model." Communications in Statistics-Theory and Methods 21.1 (1992): 89-109.

Wang, Weizhen, and Felix Famoye. "Modeling household fertility decisions with generalized Poisson regression." Journal of Population Economics 10.3 (1997): 273-283.

Yang, Zhao, James W. Hardin, and Cheryl L. Addy. "A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model." Journal of Statistical Planning and Inference 139.4 (2009): 1514-1521.

Harris, Tammy, Zhao Yang, and James W. Hardin. "Modeling underdispersed count data with generalized Poisson regression." Stata Journal 12.4 (2012): 736-747.

Examples

dgp2(1, mu = 0.75, alpha = 0.3)
#> [1] 0.2762245
pgp2(c(0, 1, 2, 3, 5, 7, 9, 10), mu = 0.75, alpha = 0.25)
#> [1] 0.5317515 0.8185413 0.9345470 0.9771263 0.9973058 0.9996865 0.9999634
#> [8] 0.9999875
qgp2(c(0.1, 0.3, 0.5, 0.9, 0.95), mu = 0.75, alpha = -0.2)
#> [1] 0 0 1 2 2
rgp2(30, mu = 0.75, alpha = 0.2)
#>  [1] 0 0 0 0 0 0 0 1 2 0 1 1 0 0 0 2 0 0 1 0 5 0 0 2 1 0 1 2 0 0