# skewhypMom: Calculate Moments of the Skew Hyperbolic Student... In SkewHyperbolic: The Skew Hyperbolic Student t-Distribution

## Description

This function can be used to calculate the raw moments, mu moments, central moments, and moments about any other given location for the skew hyperbolic t-distribution.

## Usage

 ```1 2``` ```skewhypMom(order, mu = 0, delta = 1, beta = 1, nu = 1, param = c(mu,delta,beta,nu), momType = "raw", about = 0) ```

## Arguments

 `order` Numeric. The order of the moment to be calculated. Not permitted to be a vector. Must be a positive integer, except for moments about 0. `mu` Location parameter mu, default is 0. `delta` Scale parameter delta, default is 1. `beta` Skewness parameter beta, default is 1. `nu` Shape parameter nu, default is 1. `param` Specifying the parameters as a vector of the form `c(mu,delta,beta,nu)`. `momType` Common types of moments to be calculated, default is `"raw"`, see Details. `about` Numeric. The point around which the moment is to be calculated, default is zero. See Details.

## Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specified, then the values specified by the vector `param` will overwrite the other ones. In addition the parameter values are examined by calling the function `skewhypCheckPars` to see if they are valid.

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

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

To calculate the moments of the skew hyperbolic t-distribution, the function first calculates the mu moments by the formula defined below, and then transforms them to any of the other types of moment by calling `momChangeAbout` in the `DistributionUtils` package.

The mu moments of the skew hyperbolic t-distribution are given by:

M_k = sum_{l = [(k+1)/2]}^{k} a_{k,l} beta^{2l - k} [delta^{2l}Gamma(nu/2 - l) )/ (Gamma(nu/2) 2^{l})]

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

This formula is given in Scott, W<fc>rtz and Tran (2008). Note that the [.] part of this formula is actually equivalent to the formula for the raw moments of the inverse gamma distribution, so the function calls `gammaRawMom` in the `GeneralizedHyperbolic` package when implementing the computations.

## Value

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

## Author(s)

David Scott [email protected], Fiona Grimson

## References

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

Scott, D. J., W<fc>rtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

`skewhypCheckPars`, `skewhypMean`, `is.wholenumber`, `momRecursion`, `momChangeAbout` and `gigMom`.
 ```1 2 3 4 5 6 7 8 9``` ```param = c(1,2,3,10) ##Raw moments of the skew hyperbolic t distribution skewhypMom(3, param = param, momType = "raw") ##Mu moments skewhypMom(3, param = param, momType = "mu") ##Central moments skewhypMom(3, param = param, momType = "central") ##Moments about any location skewhypMom(3, param = param, about = 5) ```