nigMeanVarMode function

Moments and Mode of the Normal Inverse Gaussian Distribution

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

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.

Returns

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.

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.

References

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

Author(s)

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

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)