Specific Normal Inverse Gaussian Distribution Moments and Mode {GeneralizedHyperbolic}

Moments and Mode of the Normal Inverse Gaussian Distribution

Description

Functions to calculate the mean, variance, skewness, kurtosis and mode of a specific normal inverse Gaussian distribution.

Usage

nigMean(mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))
nigVar(mu = 0, delta = 1, alpha = 1, beta = 0,
       param = c(mu, delta, alpha, beta))
nigSkew(mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))
nigKurt(mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))
nigMode(mu = 0, delta = 1, alpha = 1, beta = 0,
        param = c(mu, delta, alpha, beta))

Arguments

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 of the normal inverse Gaussian distribution.

Details

The mean, variance, skewness, kurtosis and mode for the normal inverse Gaussian distribution can be obtained from the functions for the generalized hyperbolic distribution as special cases (i.e., \lambda = -1/2). Likewise other moments can be obtained from the function ghypMom which implements a recursive method to moments of any desired order.

The proper formulae for the mean, variance and skewness of the normal inverse Gaussian distribution can be found in Paolella, Marc S. (2007), Chapter 9, p325.

Value

nigMean gives the mean of the normal inverse Gaussian distribution, nigVar the variance, nigSkew the skewness, nigKurt the kurtosis and nigMode the mode.

Note that the kurtosis is the standardised fourth cumulant or what is sometimes called the kurtosis excess. (See http://mathworld.wolfram.com/Kurtosis.html for a discussion.)

The parameterization of the normal inverse Gaussian distribution used for this and other components of the GeneralizedHyperbolic package is the (\alpha, \beta) one. See hyperbChangePars to transfer between parameterizations.

Author(s)

David Scott [e-mail], Christine Yang Dong

References

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

See Also

dnig, hyperbChangePars, besselK, ghypMom, ghypMean, ghypVar, ghypSkew, ghypKurt

Examples

param <- c(2, 2, 2, 1)
nigMean(param = param)
nigVar(param = param)
nigSkew(param = param)
nigKurt(param = param)
nigMode(param = param)