Generalized Poisson Version 1 Distribution

These functions provide the density function, distribution function, quantile function, and random number generation for the Generalized Poisson Version 1 (GP-1) Distribution

Usage

dgp1(x, mu = 1, phi = 1, log = FALSE)

pgp1(q, mu = 1, phi = 1, lower.tail = TRUE, log.p = FALSE)

qgp1(p, mu = 1, phi = 1, lower.tail = TRUE, log.p = FALSE)

rgp1(n, mu = 1, phi = 1)

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).

phi

single value or vector of values for the scale parameter of the distribution (the values have to be greater than -1).

log

logical; if TRUE, probabilities p 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

dgp1 gives the density, pgp1 gives the distribution function, qgp1 gives the quantile function, and rgp1 generates random deviates.

The length of the result is determined by n for rgp1, and is the maximum of the lengths of the numerical arguments for the other functions.

Details

dgp1 computes the density (PDF) of the Generalized Poisson Distribution.

pgp1 computes the CDF of the Generalized Poisson Distribution.

qgp1 computes the quantile function of the Generalized Poisson Distribution.

rgp1 generates random numbers from the Generalized Poisson Distribution.

The compound Probability Mass Function (PMF) for the Generalized Poisson distribution, version 1, (GP-1) is: $$ f(y|\phi,\mu)=\frac{\mu(\mu+\phi y)^{y-1} exp\left(-\frac{\mu+\phi y} {1+\phi}\right)}{(1+\phi)^y y!} $$

Where \(\phi\) is a scale parameter with the restriction that \(\mu>0\) is the mean value, and \(y\) is a non-negative integer. This formulation uses the mean directly.

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

If \(\phi>0\), the distribution is overdispersed. If \(\phi=0\), the distribution is equidispersed. If \(\phi<0\), the distribution is underdispersed.

Furthermore, \(\phi>-1\) is required for this distribution. When \(\phi<0\), there is also a maximum value of support for the integar value \(y\). This is \(y_max=\left\lfloor -\frac{\mu}{\phi} \right\rfloor\).

References

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

Zamani, Hossein, and Noriszura Ismail. "Functional form for the generalized Poisson regression model." Communications in Statistics-Theory and Methods 41.20 (2012): 3666-3675.

Examples

dgp1(1, mu=0.75, phi=-0.1)
#> [1] 0.4047265
pgp1(c(0,1,2,3,5,7,9,10), mu=0.75, phi=3)
#> [1] 0.8290291 0.9024552 0.9317198 0.9479434 0.9657091 0.9753198 0.9813291
#> [8] 0.9835507
qgp1(c(0.1,0.3,0.5,0.9,0.95), mu=0.75, phi=0.5)
#> [1] 0 0 0 2 3
rgp1(30, mu=0.75, phi=1.5)
#>  [1]  0  0  0  0  0  1  1 11  6  0  0  0  0  0  0  1  0  0  0  0  1  0  0  0  0
#> [26]  0  4  0  0  0