Moments and Mode of the Variance Gamma Distribution
Functions to calculate the mean, variance, skewness, kurtosis and mode of a specific variance gamma distribution.
vgMean(vgC = 0, sigma = 1, theta = 0, nu = 1, param = c(vgC,sigma,theta,nu)) vgVar(vgC = 0, sigma = 1, theta = 0, nu = 1, param = c(vgC,sigma,theta,nu)) vgSkew(vgC = 0, sigma = 1, theta = 0, nu = 1, param = c(vgC,sigma,theta,nu)) vgKurt(vgC = 0, sigma = 1, theta = 0, nu = 1, param = c(vgC,sigma,theta,nu)) vgMode(vgC = 0, sigma = 1, theta = 0, nu = 1, param = c(vgC,sigma,theta,nu))
vgC |
The location parameter c, default is equal to 0. |
sigma |
The spread parameter sigma, default is equal to 1, must be positive. |
theta |
The asymmetry parameter theta, default is equal to 0. |
nu |
The shape parameter nu, default is equal to 1, must be positive. |
param |
Specifying the parameters as a vector which takes the form
|
vgMean gives the mean of the variance gamma distribution,
vgVar the variance, vgSkew the skewness, vgKurt the
kurtosis, and vgMode the mode.
The formulae used for the mean and variance are as given in
Seneta (2004).
If nu is greater than or equal to 2, the mode is equal to the value
of the parameter c. Otherwise, it is found by a numerical
optimisation using optim.
The parameterisation of the variance gamma distribution used
for these functions is the (c,sigma,theta,nu)
one. See vgChangePars to transfer between parameterisations.
David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz
Seneta, E. (2004). Fitting the variance-gamma model to financial data. J. Appl. Prob., 41A:177–187. Kotz, S, Kozubowski, T. J., and Podgórski, K. (2001). The Laplace Distribution and Generalizations. Birkhauser, Boston, 349 p.
param <- c(2,2,2,0.5) vgMean(param = param) ## Or to specify parameter values individually, use: vgMean (2,2,2,0.5) vgVar(param = param) vgSkew(param = param) vgKurt(param = param) vgMode(param = param) maxDens <- dvg(vgMode(param = param), param = param) vgRange <- vgCalcRange(param = param, tol = 10^(-2)*maxDens) curve(dvg(x, param = param), vgRange[1], vgRange[2]) abline(v = vgMode(param = param), col = "blue") abline(v = vgMean(param = param), col = "red")
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