PoissonInverseGaussian {actuar} R Documentation

## The Poisson-Inverse Gaussian Distribution

### Description

Density function, distribution function, quantile function and random generation for the Poisson-inverse Gaussian discrete distribution with parameters `mean` and `shape`.

### Usage

```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)
```

### Arguments

 `x` vector of (positive integer) quantiles. `q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `mean, shape` parameters. Must be strictly positive. Infinite values are supported. `dispersion` an alternative way to specify the shape. `log, log.p` logical; if `TRUE`, probabilities p are returned as log(p). `lower.tail` logical; if `TRUE` (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Details

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.

### Value

`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.

### Note

`[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.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

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.

### See Also

`dpois` for the Poisson distribution, `dinvgauss` for the inverse Gaussian distribution.

### Examples

```## 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"))
```

[Package actuar version 3.1-4 Index]