NIG: Normal Inverse Gaussian Distribution

Description

Density function, distribution function, quantiles and random number generation for the normal inverse Gaussian distribution with parameter vector param. Utility routines are included for the derivative of the density function and to find suitable break points for use in determining the distribution function.

Usage

dnig(x, mu = 0, delta = 1, alpha = 1, beta = 0,
     param = c(mu, delta, alpha, beta))
pnig(q, mu = 0, delta = 1, alpha = 1, beta = 0,
     param = c(mu, delta, alpha, beta),
     lower.tail = TRUE, subdivisions = 100,
     intTol = .Machine$double.eps^0.25, valueOnly = TRUE, ...)
qnig(p, mu = 0, delta = 1, alpha = 1, beta = 0,
     param = c(mu, delta, alpha, beta),
     lower.tail = TRUE, method = c("spline","integrate"),
     nInterpol = 501, uniTol = .Machine$double.eps^0.25,
     subdivisions = 100, intTol = uniTol, ...)
rnig(n, mu = 0, delta = 1, alpha = 1, beta = 0,
     param = c(mu, delta, alpha, beta))
ddnig(x, mu = 0, delta = 1, alpha = 1, beta = 0,
     param = c(mu, delta, alpha, beta))

Value

dnig gives the density, pnig gives the distribution function, qnig gives the quantile function and rnig

generates random variates. An estimate of the accuracy of the approximation to the distribution function may be found by setting

accuracy = TRUE in the call to pnig which then returns a list with components value and error.

ddnig gives the derivative of dnig.

Arguments

x,q: Vector of quantiles.
p: Vector of probabilities.
n: Number of observations to be generated.
mu: $\mu$ is the location parameter. By default this is set to 0.
delta: $\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha: $\alpha$ is the tail parameter, with a default value of 1.
beta: $\beta$ is the skewness parameter, by default this is 0.
param: Parameter vector taking the form c(mu, delta, alpha, beta).
method: Character. If "spline" quantiles are found from a spline approximation to the distribution function. If "integrate", the distribution function used is always obtained by integration.
lower.tail: Logical. If lower.tail = TRUE, the cumulative density is taken from the lower tail.
subdivisions: The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation.
intTol: Value of rel.tol and hence abs.tol in calls to integrate. See integrate.
valueOnly: Logical. If valueOnly = TRUE calls to pghyp only return the value obtained for the integral. If valueOnly = FALSE an estimate of the accuracy of the numerical integration is also returned.
nInterpol: Number of points used in qghyp for cubic spline interpolation of the distribution function.
uniTol: Value of tol in calls to uniroot. See uniroot.
...: Passes arguments to uniroot. See Details.

Author

David Scott d.scott@auckland.ac.nz, Christine Yang Dong

Details

The normal inverse Gaussian distribution has density $$e^{\delta \sqrt{\alpha^2 - \beta^2}}% \frac{\alpha \delta}{\pi \sqrt{\delta^2 + (x - \mu)^2}}% K_1(\alpha \sqrt{\delta^2 + (x - \mu)^2})% e^{\beta (x - \mu)}$$

where $K_1()$ is the modified Bessel function of the third kind with order 1.

A succinct description of the normal inverse Gaussian distribution is given in Paolella(2007). Because both of the normal inverse Gaussian distribution and the hyperbolic distribution are special cases of the generalized hyperbolic distribution (with different values of $\lambda$), the normal inverse Gaussian distribution has the same sets of parameterizations as the hyperbolic distribution. And therefore one can use hyperbChangePars to interchange between different parameterizations for the normal inverse Gaussian distribution as well (see hyperbChangePars for details).

Each of the functions are wrapper functions for their equivalent generalized hyperbolic distribution. For example, dnig calls dghyp.

pnig breaks the real line into eight regions in order to determine the integral of dnig. The break points determining the regions are found by nigBreaks, based on the values of small, tiny, and deriv. In the extreme tails of the distribution where the probability is tiny according to nigCalcRange, the probability is taken to be zero. In the range between where the probability is tiny and small according to nigCalcRange, an exponential approximation to the hyperbolic distribution is used. In the inner part of the distribution, the range is divided in 4 regions, 2 above the mode, and 2 below. On each side of the mode, the break point which forms the 2 regions is where the derivative of the density function is deriv times the maximum value of the derivative on that side of the mode. In each of the 4 inner regions the numerical integration routine safeIntegrate (which is a wrapper for integrate) is used to integrate the density dnig.

qnig uses the breakup of the real line into the same 8 regions as pnig. For quantiles which fall in the 2 extreme regions, the quantile is returned as -Inf or Inf as appropriate. In the range between where the probability is tiny and small according to nigCalcRange, an exponential approximation to the hyperbolic distribution is used from which the quantile may be found in closed form. In the 4 inner regions splinefun is used to fit values of the distribution function generated by pnig. The quantiles are then found using the uniroot function.

pnig and qnig may generally be expected to be accurate to 5 decimal places.

Recall that the normal inverse Gaussian distribution is a special case of the generalized hyperbolic distribution and the generalized hyperbolic distribution can be represented as a particular mixture of the normal distribution where the mixing distribution is the generalized inverse Gaussian. rnig uses this representation to generate observations from the normal inverse Gaussian distribution. Generalized inverse Gaussian observations are obtained via the algorithm of Dagpunar (1989).

References

Barndorff-Nielsen, O. and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700--707. New York: Wiley.

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

Examples

Run this code

param <- c(0, 2, 1, 0)
nigRange <- nigCalcRange(param = param, tol = 10^(-3))
par(mfrow = c(1, 2))
curve(dnig(x, param = param), from = nigRange[1], to = nigRange[2],
      n = 1000)
title("Density of the\n Normal Inverse Gaussian Distribution")
curve(pnig(x, param = param), from = nigRange[1], to = nigRange[2],
      n = 1000)
title("Distribution Function of the\n Normal Inverse Gaussian Distribution")
dataVector <- rnig(500, param = param)
curve(dnig(x, param = param), range(dataVector)[1], range(dataVector)[2],
      n = 500)
hist(dataVector, freq = FALSE, add =TRUE)
title("Density and Histogram\n of the Normal Inverse Gaussian Distribution")
DistributionUtils::logHist(dataVector, main =
        "Log-Density and Log-Histogram\n of the Normal Inverse Gaussian Distribution")
curve(log(dnig(x, param = param)), add = TRUE,
      range(dataVector)[1], range(dataVector)[2], n = 500)
par(mfrow = c(2, 1))
curve(dnig(x, param = param), from = nigRange[1], to = nigRange[2],
      n = 1000)
title("Density of the\n Normal Inverse Gaussian Distribution")
curve(ddnig(x, param = param), from = nigRange[1], to = nigRange[2],
      n = 1000)
title("Derivative of the Density\n of the Normal Inverse Gaussian Distribution")

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