Moments and Mode of the Variance Gamma Distribution

Description

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

Usage

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))

Arguments

Value

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.

Author(s)

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

References

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.

See Also

dvg, vgChangePars,vgCalcRange, besselK.

Examples

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")

The Variance Gamma Distribution

Description

Density function, distribution function, quantiles and random number generation for the variance gamma distribution with parameters c (location), \sigma (spread), \theta (asymmetry) and \nu (shape). 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

  dvg(x, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), log = FALSE,
    tolerance = .Machine$double.eps ^ 0.5, ...)
  pvg(q, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), lower.tail = TRUE, log.p = FALSE,
    small = 10^(-6), tiny = 10^(-10), deriv = 0.3, subdivisions = 100,
    accuracy = FALSE, ...)
  qvg(p, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), lower.tail = TRUE, log.p = FALSE,
    small = 10^(-6), tiny = 10^(-10), deriv = 0.3, nInterpol = 100,
    subdivisions = 100, ...)
  rvg(n, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu))
  ddvg (x,  vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), log = FALSE,
    tolerance = .Machine$double.eps ^ 0.5, ...)
  vgBreaks (vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), small = 10^(-6), tiny = 10^(-10),
    deriv = 0.3, ...)

Arguments

Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specifed but with different values, then the values specified by vector param will always overwrite the other ones.

The variance gamma distribution has density

f(x)=c(c,\sigma,\theta,\nu)\times% {e^{[\theta(x-c)/\sigma^2]}}% {|x-c|^{1/\nu-1/2}}% {K_{1/\nu-1/2}}% \left(\frac{|x-c|\sqrt{2\sigma^2/\nu+\theta^2}}% {\sigma^2}\right)

where K_\nu() is the modified Bessel function of the third kind of order \nu, and

c(c,\sigma,\theta,\nu)=% \frac{2}% {\sigma\sqrt{2\pi}\nu^{1/\nu}\Gamma(1/\nu)}% \left(\frac{1}% {\sqrt{2\sigma^2/\nu+\theta^2}}\right) ^{1/\nu-1/2}

Special cases:

1. If \nu < 2 and x = c, then the density function is approximate to

f(x)= \frac{\Gamma(1/\nu-1/2)}% {\sigma\sqrt{2\pi}\nu^{1/\nu}\Gamma(1/\nu)}% \left(\frac{2\sigma^2}% {\sqrt{2\sigma^2/\nu+\theta^2}}\right) ^{1/\nu-1/2}

2. If \nu\geq2 and x = c, then the density function is taken the value Inf.

Use vgChangePars to convert from the (\mu, \sigma, \theta, \tau), or (\theta, \sigma, \kappa, \tau) parameterisations given in Kotz et al. (2001) to the (c, \sigma, \theta, \nu) parameterisation used above.

pvg breaks the real line into eight regions in order to determine the integral of dvg. The break points determining the regions are found by vgBreaks, based on the values of small, tiny, and deriv. In the extreme tails of the distribution where the probability is tiny according to vgCalcRange, the probability is taken to be zero. In the inner part of the distribution, the range is divided in 6 regions, 3 above the mode, and 3 below. On each side of the mode, there are two break points giving the required three regions. The outer break point is where the probability in the tail has the value given by the variable small. The inner break point 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 6 inner regions the numerical integration routine safeIntegrate (which is a wrapper for integrate) is used to integrate the density dvg.

qvg uses the breakup of the real line into the same 8 regions as pvg. For quantiles which fall in the 2 extreme regions, the quantile is returned as -Inf or Inf as appropriate. In the 6 inner regions splinefun is used to fit values of the distribution function generated by pvg. The quantiles are then found using the uniroot function.

pvg and qvg may generally be expected to be accurate to 5 decimal places.

The variance gamma distribution is discussed in Kotz et al (2001). It can be seen to be the weighted difference of two i.i.d. gamma variables shifted by the value of \theta. rvg uses this representation to generate oberservations from the variance gamma distribution.

Value

dvg gives the density function, pvg gives the distribution function, qvg gives the quantile function and rvg 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 pvg which then returns a list with components value and error.

ddvg gives the derivative of dvg.

vgBreaks returns a list with components:

Author(s)

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

References

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.

See Also

vgChangePars, vgCalcRange

Examples

## Use the following rules for vgCalcRange when plotting graphs for dvg,
## ddvg and pvg.
## if nu < 2, use:
##   maxDens <- dvg(vgMode(param = c(vgC, sigma, theta, nu)),
##   param = c(vgC, sigma, theta, nu), log = FALSE)
##   vgRange <- vgCalcRange(param = c(vgC, sigma, theta, nu),
##     tol = 10^(-2)*maxDens, density = TRUE)

## if nu >= 2 and theta < 0, use:
##   vgRange <- c(vgC-2,vgC+6)
## if nu >= 2 and theta > 0, use:
##   vgRange <- c(vgC-6,vgC+2)
## if nu >= 2 and theta = 0, use:
##   vgRange <- c(vgC-4,vgC+4)

# Example 1 (nu < 2)
## For dvg and pvg
param <- c(0,0.5,0,0.5)
maxDens <- dvg(vgMode(param = param), param = param, log = FALSE)
## Or to specify parameter values individually, use:
maxDens <- dvg(vgMode(0,0.5,0,0.5), 0,0.5,0,0.5, log = FALSE)

vgRange <- vgCalcRange(param = param, tol = 10^(-2)*maxDens, density = TRUE)
par(mfrow = c(1,2))
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2], n = 1000)
title("Density of the Variance Gamma Distribution")
curve(pvg(x, param = param), from = vgRange[1], to = vgRange[2], n = 1000)
title("Distribution Function of the Variance Gamma Distribution")

## For rvg
require(DistributionUtils)
dataVector <- rvg(500, param = param)
curve(dvg(x, param = param), range(dataVector)[1], range(dataVector)[2],
      n = 500)
hist(dataVector, freq = FALSE, add = TRUE)
title("Density and Histogram of the Variance Gamma Distribution")
logHist(dataVector, main =
   "Log-Density and Log-Histogram of the Generalized Hyperbolic Distribution")
curve(log(dvg(x, param = param)), add = TRUE,
      range(dataVector)[1], range(dataVector)[2], n = 500)

## For dvg and ddvg
par(mfrow = c(2,1))
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Density of the Variance Gamma Distribution")
curve(ddvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Derivative of the Density of the Variance Gamma Distribution")

# Example 2 (nu > 2 and theta = 0)
## For dvg and pvg
param <- c(0,0.5,0,3)
vgRange <- c(0-4,0+4)
par(mfrow = c(1,2))
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Density of the Variance Gamma Distribution")
curve(pvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Distribution Function of the Variance Gamma Distribution")

## For rvg
X2 <- rvg(500, param = param)
curve(dvg(x, param = param), min(X2), max(X2), n = 500)
hist(X2, freq = FALSE, add =TRUE)
title("Density and Histogram of the Variance Gamma Distribution")
DistributionUtils::logHist(X2, main =
   "Log-Density and Log-Histogramof the Generalized Hyperbolic Distribution")
curve(log(dvg(x, param = param)), add = TRUE, min(X2), max(X2), n = 500)

## For dvg and ddvg
par(mfrow = c(2,1))
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Density of the Variance Gamma Distribution")
curve(ddvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
title("Derivative of the Density of the Variance Gamma Distribution")



## Use the following rules for vgCalcRange when plotting graphs for vgBreaks.
## if (nu < 2), use:
##   maxDens <- dvg(vgMode(param =c(vgC, sigma, theta, nu)),
##     param = c(vgC, sigma, theta, nu), log = FALSE)
##   vgRange <- vgCalcRange(param = param, tol = 10^(-6)*maxDens, density = TRUE)
## if (nu >= 2) and theta < 0, use:
##    vgRange <- c(vgC-2,vgC+6)
## if (nu >= 2) and theta > 0, use:
##    vgRange <- c(vgC-6,vgC+2)
## if (nu >= 2) and theta = 0, use:
##    vgRange <- c(vgC-4,vgC+4)

## Example 3 (nu < 2)
## For vgBreaks
param <- c(0,0.5,0,0.5)
maxDens <- dvg(vgMode(param = param), param = param, log = FALSE)
vgRange <- vgCalcRange(param = param, tol = 10^(-6)*maxDens, density = TRUE)
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
bks <- vgBreaks(param = param)
abline(v = bks)
title("Density of the Variance Gamma Distribution with breaks")

## Example 4 (nu > 2 and theta = 0)
## For vgBreaks
param <- c(0,0.5,0,3)
vgRange <- c(0-4,0+4)
curve(dvg(x, param = param), from = vgRange[1], to = vgRange[2],
      n = 1000)
bks <- vgBreaks(param = param)
abline(v = bks)
title("Density of the Variance Gamma Distribution with breaks")

Variance Gamma Quantile-Quantile and Percent-Percent Plots

Description

qqvg produces a variance gamma Q-Q plot of the values in y.

ppvg produces a variance gamma P-P (percent-percent) or probability plot of the values in y. Graphical parameters may be given as arguments to qqvg and ppvg.

Usage

  qqvg(y, vgC = NULL, sigma = NULL, theta = NULL, nu = NULL,
    param = c(vgC, sigma, theta, nu), main = "Variance Gamma Q-Q Plot",
    xlab = "Theoretical Quantiles", ylab = "Sample Quantiles",
    plot.it = TRUE, line = TRUE, ...)

  ppvg(y, vgC = NULL, sigma = NULL, theta = NULL, nu = NULL,
    param = c(vgC, sigma, theta, nu), main = "Variance Gamma P-P Plot",
    xlab = "Uniform Quantiles",
    ylab = "Probability-integral-transformed Data", plot.it = TRUE,
    line = TRUE, ...)

Arguments

Details

Users may specify the parameter values of the data sample y using argument param. If param is not specified by users, then the values are estimated from y by vgFit. For more details of fiting a variance gamma distribution to data, see vgFit.

Value

For qqvg and ppvg, a list with components:

Author(s)

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

References

Wilk, M. B. and Gnanadesikan, R. (1968) Probability plotting methods for the analysis of data. Biometrika. 55, 1–17.

See Also

ppoints, dvg.

Examples

## Example 1: the parameter values are known
par(mfrow = c(1,2))
y <- rvg(200, param = c(2,2,1,2))
qqvg(y, param = c(2,2,1,2),line = FALSE)
abline(0, 1, col = 2)
ppvg(y, param = c(2,2,1,2))

## Example 2: the parameter values are unknown
par(mfrow = c(1,2))
y <- rvg(200, param = c(2,2,1,2))
qqvg(y, line = FALSE)
abline(0, 1, col = 2)
ppvg(y)

Summarizing Variance Gamma Distribution Fit

Description

summary Method for class "vgFit".

Usage

## S3 method for class 'vgFit'
summary(object, ...)

## S3 method for class 'summary.vgFit'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Details

summary.vgFit calculates standard errors for the estimates of c, \sigma, \theta, and \nu of the variance gamma distribution parameter vector param if the Hessian from the call to optim or nlm is available. Because the parameters in the call to the optimiser are c, \log(\sigma), \theta and \log(\nu), the delta method is used to obtain the standard errors for \sigma and \nu.

Value

If the Hessian is available, summary.vgFit computes standard errors for the estimates of c, \sigma, \theta, and \nu, and adds them to object as object$sds. Otherwise, no calculations are performed and the composition of object is unaltered.

summary.vgFit invisibly returns x with class changed to summary.vgFit.

See vgFit for the composition of an object of class vgFit.

print.summary.vgFit prints a summary in the same format as print.vgFit when the Hessian is not available from the fit. When the Hessian is available, the standard errors for the parameter estimates are printed in parentheses beneath the parameter estimates, in the manner of fitdistr in the package MASS.

See Also

vgFit, summary.

Examples

### Continuing the  vgFit(.) example:
param <- c(0,0.5,0,0.5)
dataVector <- rvg(500, param = param)
fit <- vgFit(dataVector)
print(fit)
summary(fit)          

Range of a Variance Gamma Distribution

Description

Given the parameter vector param or the idividual parameter values (c,\sigma,\theta,\nu) of a variance gamma distribution, this function determines the range outside of which the density function is negligible, to a specified tolerance. The parameterization used is the (c,\sigma,\theta,\nu) one (see dvg). To use another parameterization, use vgChangePars.

Usage

  vgCalcRange(vgC = 0, sigma = 1, theta = 0, nu = 1, 
    param = c(vgC, sigma, theta, nu), tol = 10^(-5), density = TRUE, ...)

Arguments

Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specifed but with different values, then the values specified by vector param will always overwrite the other ones.

The particular variance gamma distribution being considered is specified by the value of the parameter param.

If density = TRUE, the function gives a range, outside of which the density is less than the given tolerance. Useful for plotting the density. Also used in determining break points for the separate sections over which numerical integration is used to determine the distribution function. The points are found by using uniroot on the density function.

If density = FALSE, the function returns the message: "Distribution function bounds not yet implemented".

Value

A two-component vector giving the lower and upper ends of the range.

Author(s)

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

References

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.

See Also

dvg, vgChangePars

Examples

## Use the following rules for vgCalcRange when plotting graphs for dvg,
## ddvg and pvg.
## if nu < 2, use:
##   maxDens <- dvg(vgMode(param = c(vgC, sigma, theta, nu)),
##   param = c(vgC, sigma, theta, nu), log = FALSE)
##   vgRange <- vgCalcRange(param = c(vgC, sigma, theta, nu),
##     tol = 10^(-2)*maxDens, density = TRUE)

## if nu >= 2 and theta < 0, use:
##   vgRange <- c(vgC-2,vgC+6)
## if nu >= 2 and theta > 0, use:
##   vgRange <- c(vgC-6,vgC+2)
## if nu >= 2 and theta = 0, use:
##   vgRange <- c(vgC-4,vgC+4)

param <- c(0,0.5,0,0.5)
maxDens <- dvg(vgMode(param = param), param = param)
vgRange <- vgCalcRange(param = param, tol = 10^(-2)*maxDens)
vgRange
curve(dvg(x, param = param), vgRange[1], vgRange[2])
curve(dvg(x, param = param), vgRange[1], vgRange[2])

param <- c(2,2,0,3)
vgRange <- c(2-4,2+4)
vgRange
curve(dvg(x, param = param), vgRange[1], vgRange[2])
## Not run: vgCalcRange(param = param, tol = 10^(-3), density = FALSE)

Change Parameterizations of the Variance Gamma Distribution

Description

This function interchanges between the following 4 parameterizations of the variance gamma distribution:

1. c, \sigma, \theta, \nu

2. \theta, \sigma, \mu, \tau

3. \theta, \sigma, \kappa, \tau

4. \lambda, \alpha, \beta, \mu

The first set of parameterizations is given in Seneta (2004). The second and third ones are the parameterizations given in Kotz et al. (2001). The last set takes the form of the generalized hyperbolic distribution parameterization. \delta is not included since the variance gamma distribution is a limiting case of generalized hyperbolic distribution with \delta always equal to 0.

Usage

vgChangePars(from, to, param, noNames = FALSE)

Arguments

Details

In the 3 parameterizations, the following must be positive:

1. \sigma, \nu

2. \sigma, \tau

3. \sigma, \tau

4. \lambda, \alpha

In addition in the 4th parameterization, the absolute value of \beta must be less than \alpha.

Value

A numerical vector of length 4 representing param in the to parameterization.

Author(s)

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

References

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.

See Also

dvg, vgMom

Examples

param1 <- c(2,2,1,3)                   # Parameterization 1
param2 <- vgChangePars(1, 2, param1)   # Convert to parameterization 2
param2                                 # Parameterization 2
vgChangePars(2, 1, as.numeric(param2)) # Convert back to parameterization 1

param3 <- c(1,2,0,0.5)                 # Parameterization 3
param1 <- vgChangePars(3, 1, param3)   # Convert to parameterization 1
param1                                 # Parameterization 1
vgChangePars(1, 3, as.numeric(param1)) # Convert back to parameterization 3

Check Parameters of the Variance Gamma Distribution

Description

Given a putative set of parameters for the variance gamma distribution, the functions checks if the parameters are in the correct range, and if the set has the correct length of 4.

Usage

  vgCheckPars(param, ...)

Arguments

Details

The vector param takes the form c(c, sigma, theta, nu). If either sigma or nu is negative, then an error message is returned.

If the vector param has a length not equal to 4, then an error message is returned.

Value

A list with components:

Author(s)

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

See Also

dvg, vgMom

Examples

 vgCheckPars(c(0,1,0,1))      # normal
 vgCheckPars(c(0,0,0,1))      # error
 vgCheckPars(c(0,1,0,-2))     # error
 vgCheckPars(c(0,1,0))        # error

Fit the Variance Gamma to Data

Description

Fits a variance gamma distribution to data. Displays the histogram, log-histogram (both with fitted densities), Q-Q plot and P-P plot for the fit which has the maximum likelihood.

Usage

  vgFit(x, freq = NULL, breaks = NULL, paramStart = NULL,
        startMethod = "Nelder-Mead", startValues = "SL",
        method = "Nelder-Mead", hessian = FALSE,
        plots = FALSE, printOut = FALSE,
        controlBFGS = list(maxit = 200),
        controlNM = list(maxit = 1000), maxitNLM = 1500, ...)


  ## S3 method for class 'vgFit'
print(x, digits = max(3, getOption("digits") - 3), ...)

  ## S3 method for class 'vgFit'
plot(x, which = 1:4,
       plotTitles = paste(c("Histogram of ","Log-Histogram of ",
                            "Q-Q Plot of ","P-P Plot of "), x$obsName,
                          sep = ""),
       ask = prod(par("mfcol")) < length(which) && dev.interactive(),
          ...)

Arguments

Details

startMethod can be either "BFGS" or "Nelder-Mead".

startValues can be one of the following:

"US"

User-supplied.

"SL"

Based on a fitted skew-Laplace distribution.

"MoM"

Method of moments.

For the details concerning the use of paramStart, startMethod, and startValues, see vgFitStart.

The three optimisation methods currently available are:

"BFGS"

Uses the quasi-Newton method "BFGS" as documented in optim.

"Nelder-Mead"

Uses an implementation of the Nelder and Mead method as documented in optim.

"nlm"

Uses the nlm function in R.

For details of how to pass control information for optimisation using optim and nlm, see optim and nlm.

When method = "Nelder-Mead" is used, very rarely, it would return an error message of "error in optim(paramStart,...)", use method = "BFGS" or method = "nlm" instead in that case.

When method = "nlm" is used, warnings may be produced. These do not appear to be a problem.

Value

A list with components:

Author(s)

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

References

Seneta, E. (2004). Fitting the variance-gamma model to financial data. J. Appl. Prob., 41A:177–187.

See Also

optim, nlm, par, hist, logHist, qqvg, ppvg, dskewlap and vgFitStart.

Examples

param <- c(0,0.5,0,0.5)
dataVector <- rvg(500, param = param)
## See how well vgFit works
vgFit(dataVector)
vgFit(dataVector, plots = TRUE)
fit <- vgFit(dataVector)
par(mfrow = c(1,2))
plot(fit, which = c(1,3))

## Use nlm instead of default
param <- c(0,0.5,0,0.5)
dataVector <- rvg(500, param = param)
vgFit(dataVector, method = "nlm", hessian = TRUE)


## Use BFGS instead of deault
param <- c(0,0.5,0,0.5)
dataVector <- rvg(500, param = param)
vgFit(dataVector, method = "BFGS", hessian = TRUE)

Find Starting Values for Fitting a Variance Gamma Distribution

Description

Finds starting values for input to a maximum likelihood routine for fitting variance gamma distribution to data.

Usage

  vgFitStart(x, breaks = NULL, startValues = "SL", paramStart = NULL,
             startMethodSL = "Nelder-Mead",
             startMethodMoM = "Nelder-Mead", ...)
  vgFitStartMoM(x, startMethodMoM = "Nelder-Mead", ...)

Arguments

Details

Possible values of the argument startValues are the following:

"US"

User-supplied.

"SL"

Based on a fitted skew-Laplace distribution.

"MoM"

Method of moments.

If startValues = "US" then a value must be supplied for paramStart.

If startValues = "MoM", vgFitStartMoM is called. These starting values are based on Barndorff-Nielsen et al (1985).

If startValues = "SL", or startValues = "MoM" an initial optimisation is needed to find the starting values. These optimisations call optim.

Value

vgFitStart returns a list with components:

vgFitStartMoM returns only the method of moments estimates as a vector with elements vgC, lSigma (log of sigma), theta and lNu (log of nu).

Author(s)

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

References

Seneta, E. (2004). Fitting the variance-gamma model to financial data. J. Appl. Prob., 41A:177–187.

See Also

dvg, dskewlap, vgFit, hist, and optim.

Examples

param <- c(0,0.5,0,0.5)
dataVector <- rvg(500, param = param)
vgFitStart(dataVector,startValues="SL")
vgFitStartMoM(dataVector)
vgFitStart(dataVector,startValues="MoM")

Calculate Moments of the Variance Gamma Distribution

Description

This function can be used to calculate raw moments, mu moments, central moments and moments about any other given location for the variance gamma (VG) distribution.

Usage

  vgMom(order, vgC = 0, sigma = 1, theta = 0, nu = 1,
    param = c(vgC,sigma,theta,nu), momType = "raw", about = 0)

Arguments

Details

For the parameters of the variance gamma distribution, users may either specify the values individually or as a vector. If both forms are specified but with different values, then the values specified by vector param will always overwrite the other ones. In addition, the parameters values are examined by calling the function vgCheckPars to see if they are valid for the VG distribution.

order is also checked by calling the function is.wholenumber in DistributionUtils package to see whether a whole number is given.

momType can be either "raw" (moments about zero), "mu" (moments about vgC), or "central" (moments about mean). If one of these moment types is specified, then there is no need to specify the about value. For moments about any other location, the about value must be specified. In the case that both momType and about are specified and contradicting, the function will always calculate the moments based on about rather than momType.

To calculate moments of the VG distribution, the function first calculates mu moments by the formula defined below and then transforms mu moments to central moments or raw moments or moments about any other location as required by calling momChangeAbout in DistributionUtils package.

To calculate mu moments of the variance gamma distribution, the function first transforms the parameterization of c,\sigma,\theta,\nu to the generalized hyperbolic distribution's parameterization of \lambda, \alpha, \beta, \mu (see vgChangePars for details). Then, the mu moments of the variance gamma distribution are given by

\sum_{\ell = \lfloor(k+1)/2\rfloor}^{k} a_{k, \ell} \beta^{2\ell - k} \lfloor\Gamma(\lambda+\ell)/\Gamma(\lambda) 2^\ell/(\alpha^2-\beta^2)^\ell\rfloor

where k = \code{order} and k > 0 and a_{k, \ell} is the recursive coefficient (see momRecursion for details).

This formula is developed from the mu moments formula of the generalized hyperbolic distribution given in Scott, Würtz and Tran (2008). Note that the part in [] of this equation is actually equivalent to the formula of raw moments of the gamma distribution. So the function calls gammaRawMom in GeneralizedHyperbolic package when implementing the computations.

Value

The moment specified. In the case of raw moments, Inf is returned if the moment is infinite.

Author(s)

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

References

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley Scott, D. J., Würtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

See Also

vgCheckPars, vgChangePars, vgMean, vgVar, vgSkew, vgKurt, is.wholenumber, momRecursion, momChangeAbout and momIntegrated.

Examples

  ### Raw moments of the VG distribution
  vgMom(3, param=c(2,1,2,1), momType = "raw")

  ### Mu moments of the VG distribution
  vgMom(2, param=c(2,1,2,1), momType = "mu")

  ### Central moments of the VG distribution
  vgMom(4, param=c(2,1,2,1), momType = "central")

  ### Moments about any locations
  vgMom(4, param=c(2,1,2,1), about = 1)

Parameter Sets for Variance Gamma Distribution

Description

These objects store different parameter sets of the Variance Gamma distribution for testing or demonstrating purpose as matrixes. Specifically, the parameter sets vgSmallShape and vgLargeShape have constant (standard) location and spread parameters of c=0 and \sigma=1; where asymmetry and shape parameters vary from \theta=(-2, 0, 2) and \nu=(0.5, 1, 2) for vgSmallShape and \theta=(-4, -2, 0, 2, 4) and \nu=(0.25, 0.5, 1, 2, 4) for vgLargeShape.

The parameter sets vgSmallParam and vgLargeParam have varied values of all 4 parameters. vgSmallParam contains all of the parameter combinations from c=(-2, 0, 2), \sigma=(0.5, 1, 2), \theta=(-2, 0, 2) and \nu=(0.5, 1, 2). vgLargeParam contains all of the parameter combinations from c=(-4, -2, 0, 2, 4), \sigma=(0.25, 0.5, 1, 2, 4), \theta=(-4, -2, 0, 2, 4) and \nu=(0.25, 0.5, 1, 2, 4).

Usage

  data(vgParam)

Format

vgSmallShape: a 9 by 4 matrix; vgLargeShape: a 25 by 4 matrix; vgSmallParam: a 81 by 4 matrix; vgLargeParam: a 625 by 4 matrix.

Author(s)

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

Examples

data(vgParam)
## Testing the accuracy of vgMean
for (i in 1:nrow(vgSmallParam)) {
    param <- vgSmallParam[i,]
    x <- rvg(10000,param = param)
    sampleMean <- mean(x)
    funMean <- vgMean(param = param)
    difference <- abs(sampleMean - funMean)
    print(difference)
}