The Poisson-Inverse Gaussian Distribution
Density function, distribution function, quantile function and random
generation for the Poisson-inverse Gaussian discrete distribution with
parameters mean and shape.
dpoisinvgauss(x, mean, shape = 1, dispersion = 1/shape,
log = FALSE)
ppoisinvgauss(q, mean, shape = 1, dispersion = 1/shape,
lower.tail = TRUE, log.p = FALSE)
qpoisinvgauss(p, mean, shape = 1, dispersion = 1/shape,
lower.tail = TRUE, log.p = FALSE)
rpoisinvgauss(n, mean, shape = 1, dispersion = 1/shape)x |
vector of (positive integer) quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
mean, shape |
parameters. Must be strictly positive. Infinite values are supported. |
dispersion |
an alternative way to specify the shape. |
log, log.p |
logical; if |
lower.tail |
logical; if |
The Poisson-inverse Gaussian distribution is the result of the continuous mixture between a Poisson distribution and an inverse Gaussian, that is, the distribution with probability mass function
p(x) = int_0^Inf (y^x exp(-y))/x! g(y; μ, φ) dy,
where g(y; μ, φ) is the density
function of the inverse Gaussian distribution with parameters
mean = μ and dispersion = φ (see
dinvgauss).
The resulting probability mass function is
p(x) = sqrt(2/(π φ)) exp(1/(φ μ))/x! * [√(2 φ (1 + 1/(2 φ μ^2)))]^(-(x-1/2)) * K(√((2/φ) (1 + 1/(2 φ μ^2))); x-1/2),
for x = 0, 1, …, μ > 0, φ > 0 and where
K(x; ν) is the modified Bessel function of the third
kind implemented by R's besselK() and defined in its
help.
The limiting case μ = Inf has well defined probability mass and distribution functions, but has no finite strictly positive, integer moments. The pmf in this case reduces to
p(x) = sqrt(2/(π φ)) 1/x! [√(2 φ)]^(-(x-1/2)) * K(√(2/φ); x-1/2).
The limiting case φ = 0 is a degenerate distribution in x = 0.
If an element of x is not integer, the result of
dpoisinvgauss is zero, with a warning.
The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.
dpoisinvgauss gives the probability mass function,
ppoisinvgauss gives the distribution function,
qpoisinvgauss gives the quantile function, and
rpoisinvgauss generates random deviates.
Invalid arguments will result in return value NaN, with a warning.
The length of the result is determined by n for
rpoisinvgauss, and is the maximum of the lengths of the
numerical arguments for the other functions.
[dpqr]pig are aliases for [dpqr]poisinvgauss.
qpoisinvgauss is based on qbinom et al.; it uses the
Cornish–Fisher Expansion to include a skewness correction to a normal
approximation, followed by a search.
Vincent Goulet vincent.goulet@act.ulaval.ca
Holla, M. S. (1966), “On a Poisson-Inverse Gaussian Distribution”, Metrika, vol. 15, p. 377-384.
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005), Univariate Discrete Distributions, Third Edition, Wiley.
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
Shaban, S. A., (1981) “Computation of the poisson-inverse gaussian distribution”, Communications in Statistics - Theory and Methods, vol. 10, no. 14, p. 1389-1399.
## Tables I and II of Shaban (1981)
x <- 0:2
sapply(c(0.4, 0.8, 1), dpoisinvgauss, x = x, mean = 0.1)
sapply(c(40, 80, 100, 130), dpoisinvgauss, x = x, mean = 1)
qpoisinvgauss(ppoisinvgauss(0:10, 1, dis = 2.5), 1, dis = 2.5)
x <- rpoisinvgauss(1000, 1, dis = 2.5)
y <- sort(unique(x))
plot(y, table(x)/length(x), type = "h", lwd = 2,
pch = 19, col = "black", xlab = "x", ylab = "p(x)",
main = "Empirical vs theoretical probabilities")
points(y, dpoisinvgauss(y, 1, dis = 2.5),
pch = 19, col = "red")
legend("topright", c("empirical", "theoretical"),
lty = c(1, NA), pch = c(NA, 19), col = c("black", "red"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.