nig: The Normal Inverse-Gaussian (NIG) Distribution

In ngme2: Linear Latent Non-Gaussian Models with Flexible Distributions

The Normal Inverse-Gaussian (NIG) Distribution

Description

Density, distribution function, quantile function and random generation for the normal inverse-Gaussian distribution with parameters p, a and b.

Usage

dnig(x, delta, mu, nu, sigma, h = NULL, log = FALSE)

rnig(n, delta, mu, nu, sigma, h = NULL, seed = 0)

pnig(q, delta, mu, nu, sigma, h = NULL, lower.tail = TRUE, log.p = FALSE)

qnig(p, delta, mu, nu, sigma, h = NULL, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q	vector of quantiles.
delta	A numeric value for the location parameter.
mu	A numeric value for the shift parameter.
nu	A numeric value for the shape parameter.
sigma	A numeric value for the scaling parameter.
h	A numeric value for the additional parameter, see details.
log, log.p	logical; if TRUE, probabilities/densities p are returned as log(p).
n	number of observations.
seed	Seed for the random generation.
lower.tail	logical; if TRUE, probabilities are P[X\leq x], otherwise, P[X>x].
p	vector of probabilities.

Details

The normal inverse-Gaussian distribution has density given by

f(x; \delta, \mu, \sigma, \nu) = \frac{e^{\nu+\mu(x-\delta)/\sigma^2}\sqrt{\nu\mu^2/\sigma^2+\nu^2}}{\pi\sqrt{\nu\sigma^2+(x-\delta)^2}} K_1(\sqrt{(\nu\sigma^2+(x-\delta)^2)(\mu^2/\sigma^4+\nu/\sigma^2)}),

where K_p is modified Bessel function of the second kind of order p, x>0, \nu>0 and \mu,\delta, \sigma\in\mathbb{R}. See Barndorff-Nielsen (1977, 1978 and 1997) for further details.

The additional parameter h is used when

V\sim IG(\nu,\nu h^{2})

. By the infinite divisibility,

\frac{1}{h} V \sim IG(\nu h, \nu h)

. Then

\delta+\mu V + \sigma \sqrt{V} Z

has the distribution of

NIG(\delta=-\mu h,\mu= \mu h, \sigma=\sigma \sqrt{h}, \nu=\nu h).

Value

dnig gives the density, pnig gives the distribution function, qnig gives the quantile function, and rnig generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rnig.

References

Barndorff-Nielsen, O. (1977) Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London.

Series A, Mathematical and Physical Sciences. The Royal Society. 353, 401–409. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1098/rspa.1977.0041")}

Barndorff-Nielsen, O. (1978) Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics. 5, 151–157.

Barndorff-Nielsen, O. (1997) Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics. 24, 1-13. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/1467-9469.00045")}

See Also

dgig, dig, digam

Examples

rnig(100, delta = 0, mu = 5, sigma = 1, nu = 1)
pnig(0.4, delta = 0, mu = 5, sigma = 1, nu = 1)
qnig(0.8, delta = 0, mu = 5, sigma = 1, nu = 1)
plot(function(x){dnig(x, delta = 0, mu = 5, sigma = 1, nu = 1)}, main =
"Normal inverse-Gaussian density", ylab = "Probability density",
xlim = c(0,10))

ngme2 documentation built on May 20, 2026, 9:10 a.m.