Poisson-Inverse-Gaussian Distribution

These functions provide the density function, distribution function, quantile function, and random number generation for the Poisson-Inverse-Gaussian (PInvGaus) Distribution.

These functions provide the density function, distribution function, quantile function, and random number generation for the Poisson-Inverse-Gaussian (PInvGaus) Distribution

Usage

dpinvgaus(x, mu = 1, eta = 1, form = "Type 1", log = FALSE)

ppinvgaus(
  q,
  mu = 1,
  eta = 1,
  form = "Type 1",
  lower.tail = TRUE,
  log.p = FALSE
)

qpinvgaus(p, mu = 1, eta = 1, form = "Type 1")

rpinvgaus(n, mu = 1, eta = 1, form = "Type 1")

dpinvgaus(x, mu = 1, eta = 1, form = "Type 1", log = FALSE)

ppinvgaus(
  q,
  mu = 1,
  eta = 1,
  form = "Type 1",
  lower.tail = TRUE,
  log.p = FALSE
)

qpinvgaus(p, mu = 1, eta = 1, form = "Type 1")

rpinvgaus(n, mu = 1, eta = 1, form = "Type 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).

eta

single value or vector of values for the scale parameter of the distribution (the values have to be greater than 0).

form

optional parameter indicating which formulation to use. Options include "Type 1" which is the standard form or "Type 2" which follows the formulation by Dean et. al. (1987).

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.

Details

The Poisson-Inverse-Gaussian distribution is a special case of the Sichel distribution (Cameron & Trivedi, 2013). It is also known as a univariate Sichel distribution (Hilbe, 2011).

dpinvgaus computes the PDF of the Poisson-Inverse-Gaussian dist.

ppinvgaus computes the CDF of the Poisson-Inverse-Gaussian dist.

qpinvgaus computes quantiles of the Poisson-Inverse-Gaussian dist.

rpinvgaus generates random numbers from the distribution.

The PMF (Type 1) is: $$ f(y|\eta,\mu)=\begin{cases} f(0)=\exp\left(\frac{\mu}{\eta}(1-\sqrt{1+2\eta})\right)\\ f(y>0)=f(0)\frac{\mu^y}{y!}(1+2\eta)^{-y/2} \sum_{j=0}^{y-1}\frac{\Gamma(y+j)} {\Gamma(y-j)\Gamma(j+1)} \left(\frac{\eta}{2\mu}\right)^j(1+2\eta)^{-j/2} \end{cases}$$

The variance is: $$\sigma^2=\mu+\eta\mu$$

Type 2 modifies \(\eta\) → \(\eta\mu\): $$ f(0)=\exp\left(\frac{1}{\eta}(1-\sqrt{1+2\eta\mu})\right) $$ $$ f(y>0)=f(0)\frac{\mu^y}{y!}(1+2\eta\mu)^{-y/2} \sum_{j=0}^{y-1}\frac{\Gamma(y+j)} {\Gamma(y-j)\Gamma(j+1)} \left(\frac{\eta}{2}\right)^j(1+2\eta\mu)^{-j/2} $$

Resulting variance: $$\sigma^2=\mu+\eta\mu^2$$

dpinvgaus gives the density, ppinvgaus gives the distribution function, qpinvgaus gives the quantile function, and rpinvgaus generates random deviates.

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

The Poisson-Inverse-Gaussian distribution is a special case of the Sichel distribution, as noted by Cameron & Trivedi (2013). It is also known as a univariate Sichel distribution (Hilbe, 2011).

dpinvgaus computes the density (PDF) of the Poisson-Inverse-Gaussian Distribution.

ppinvgaus computes the CDF of the Poisson-Inverse-Gaussian Distribution.

qpinvgaus computes the quantile function of the Poisson-Inverse-Gaussian Distribution.

rpinvgaus generates random numbers from the Poisson-Inverse-Gamma Distribution.

The compound Probability Mass Function (PMF) for the Poisson-Inverse-Gaussian distribution (Type 1) is (Cameron & Trivedi, 2013): $$f(y|\eta,\mu) = \begin{cases} f(y = 0) = \exp \left( \frac{\mu}{\eta} \left(1-\sqrt{1 + 2\eta}\right)\right) \\ f(y|y>0) = f(y = 0)\frac{\mu^y}{y!}(1 + 2\eta)^{-y / 2} \cdot \sum_{j=0}^{y-1} \frac{\Gamma(y+j)}{\Gamma(y-j)\Gamma(j+1)} \left( \frac{\eta}{2\mu}\right)^2(1 + 2\eta)^{-j / 2} \end{cases}$$

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

The variance of the distribution is: $$\sigma^2=\mu+\eta\mu$$

The alternative parametrization by Dean et. al. (1987) replaces \(\eta\) with \(\eta\mu\). This version (Type 2) has the PMF: $$f(y|\eta,\mu)=\begin{cases} f(y=0) = \exp \left(\frac{1}{\eta} \left(1-\sqrt{1+2\eta\mu}\right) \right) \\ f(y|y > 0) = f(y=0) \frac{\mu^y}{y!} (1+2\eta\mu)^{-y/2} \cdot \sum_{j=0}^{y-1} \frac{\Gamma(y+j)}{\Gamma(y-j)\Gamma(j+1)} \left(\frac{\eta}{2} \right)^2(1+2\eta\mu)^{-j/2} \end{cases}$$

This results in the variance of: $$\sigma^2=\mu+\eta\mu^2$$

References

Cameron & Trivedi (2013). Regression Analysis of Count Data. Dean, Lawless & Willmot (1989). Mixed Poisson–Inverse Gaussian Models. Hilbe (2011). Negative Binomial Regression.

Cameron, A. C., & Trivedi, P. K. (2013). Regression analysis of count data, 2nd Edition. Cambridge university press.

Dean, C., Lawless, J. F., & Willmot, G. E. (1989). A mixed Poisson–Inverse‐Gaussian regression model. Canadian Journal of Statistics, 17(2), 171-181.

Hilbe, J. M. (2011). Negative binomial regression. Cambridge University Press.

Examples

dpinvgaus(1, mu=0.75, eta=1)
#> [1] 0.250067
ppinvgaus(c(0,1,2,3,5,7,9,10), mu=0.75, eta=3, form="Type 2")
#> [1] 0.6386475 0.8428877 0.9173222 0.9512540 0.9797144 0.9904002 0.9951184
#> [8] 0.9964543
qpinvgaus(c(0.1,0.3,0.5,0.9,0.95), mu=0.75, eta=0.5, form="Type 2")
#> [1] 0 0 0 2 3
rpinvgaus(30, mu=0.75, eta=1.5)
#>  [1] 3 1 0 0 0 1 6 0 0 0 0 0 1 0 4 0 2 2 0 0 0 0 1 0 0 0 2 3 0 0

dpinvgaus(1, mu=0.75, eta=1)
#> [1] 0.250067
ppinvgaus(c(0,1,2,3,5,7,9,10), mu=0.75, eta=3, form="Type 2")
#> [1] 0.6386475 0.8428877 0.9173222 0.9512540 0.9797144 0.9904002 0.9951184
#> [8] 0.9964543
qpinvgaus(c(0.1,0.3,0.5,0.9,0.95), mu=0.75, eta=0.5, form="Type 2")
#> [1] 0 0 0 2 3
rpinvgaus(30, mu=0.75, eta=1.5)
#>  [1]  0  0 11  0  2  0  0  0  0  0  0  0  0  0  0  0  2  0  1  3  0  1  0  0  0
#> [26]  0  1  0  1  0