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

 Specific Hyperbolic Distribution Moments and Mode R Documentation

## Moments and Mode of the Hyperbolic Distribution

### Description

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

### Usage

```hyperbMean(Theta)
hyperbVar(Theta)
hyperbSkew(Theta)
hyperbKurt(Theta)
hyperbMode(Theta)
```

### Arguments

 `Theta` Parameter vector of the hyperbolic distribution.

### Details

The formulae used for the mean, variance and mode are as given in Barndorff-Nielsen and Bl<e6>sild (1983), p. 702. The formulae used for the skewness and kurtosis are those of Barndorff-Nielsen and Bl<e6>sild (1981), Appendix 2.

Note that the variance, skewness and kurtosis can be obtained from the functions for the generalized hyperbolic distribution as special cases. Likewise other moments can be obtained from the function `ghypMom` which implements a recursive method to moments of any desired order. Note that functions for the generalized hyperbolic distribution use a different parameterization, so care is required.

### Value

`hyperbMean` gives the mean of the hyperbolic distribution, `hyperbVar` the variance, `hyperbSkew` the skewness, `hyperbKurt` the kurtosis and `hyperbMode` 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 hyperbolic distribution used for this and other components of the `HyperbolicDist` package is the (pi,zeta) one. See `hyperbChangePars` to transfer between parameterizations.

### Author(s)

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

### References

Barndorff-Nielsen, O. and Bl<e6>sild, P (1981). Hyperbolic distributions and ramifications: contributions to theory and application. In Statistical Distributions in Scientific Work, eds., Taillie, C., Patil, G. P., and Baldessari, B. A., Vol. 4, pp. 19–44. Dordrecht: Reidel.

Barndorff-Nielsen, O. and Bl<e6>sild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

`dhyperb`, `hyperbChangePars`, `besselK`, `ghypMom`, `ghypMean`, `ghypVar`, `ghypSkew`, `ghypKurt`

### Examples

```Theta <- c(2,2,2,2)
hyperbMean(Theta)
hyperbVar(Theta)
hyperbSkew(Theta)
hyperbKurt(Theta)
hyperbMode(Theta)
```

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