# ghypMeanVarMode: Moments and Mode of the Generalized Hyperbolic Distribution In HyperbolicDist: The Hyperbolic Distribution

 Specific Generalized Hyperbolic Moments and Mode R Documentation

## Moments and Mode of the Generalized Hyperbolic Distribution

### Description

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

### Usage

```ghypMean(Theta)
ghypVar(Theta)
ghypSkew(Theta)
ghypKurt(Theta)
ghypMode(Theta)
```

### Arguments

 `Theta` Parameter vector of the generalized hyperbolic distribution.

### Value

`ghypMean` gives the mean of the generalized hyperbolic distribution, `ghypVar` the variance, `ghypSkew` the skewness, `ghypKurt` the kurtosis, and `ghypMode` the mode. The formulae used for the mean is given in Prause (1999). The variance, skewness and kurtosis are obtained using the recursive formula implemented in `ghypMom` which can calculate moments of all orders about any point.

The mode is found by a numerical optimisation using `optim`. For the special case of the hyperbolic distribution a formula for the mode is available, see `hyperbMode`.

The parameterization of the generalized hyperbolic distribution used for these functions is the (alpha,beta) one. See `ghypChangePars` to transfer between parameterizations.

### Author(s)

David Scott d.scott@auckland.ac.nz, Thomas Tran

### References

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

`dghyp`, `ghypChangePars`, `besselK`, `RLambda`.

### Examples

```Theta <- c(2,2,1,2,2)
ghypMean(Theta)
ghypVar(Theta)
ghypSkew(Theta)
ghypKurt(Theta)
ghypMode(Theta)
maxDens <- dghyp(ghypMode(Theta), Theta)
ghypRange <- ghypCalcRange(Theta, tol = 10^(-3)*maxDens)
curve(dghyp(x, Theta), ghypRange[1], ghypRange[2])
abline(v = ghypMode(Theta), col = "blue")
abline(v = ghypMean(Theta), col = "red")
```

HyperbolicDist documentation built on March 18, 2022, 6:23 p.m.