momRecursion {GeneralizedHyperbolic}

Computes the moment coefficients recursively for generalized hyperbolic and related distributions

Description

This function computes all of the moments coefficients by recursion based on Scott, Würtz and Tran (2008). See Details for the formula.

Usage

  momRecursion(order = 12, printMatrix = FALSE)

Arguments

order

Numeric. The order of the moment coefficients to be calculated. Not permitted to be a vector. Must be a positive whole number except for moments about zero.

printMatrix

Logical. Should the coefficients matrix be printed?

Details

The moment coefficients recursively as a_{1,1}=1 and

a_{k,\ell} = a_{k-1, \ell-1} + (2 \ell - k + 1) a_{k-1, \ell}

with a_{k,\ell} = 0 for \ell<\lfloor(k+1)/2\rfloor or \ell>k where k = order, \ell is equal to the integers from (k+1)/2 to k.

This formula is given in Scott, Würtz and Tran (2008, working paper).

The function also calculates M which is equal to 2\ell - k. It is a common term which will appear in the formulae for calculating moments of generalized hyperbolic and related distributions.

Value

a

The non-zero moment coefficients for the specified order.

l

Integers from (order+1)/2 to order. It is used when computing the moment coefficients and the mu moments.

M

The common term used when computing mu moments for generalized hyperbolic and related distributions, M = 2\ell - k, k=order

lmin

The minimum of \ell, which is equal to (order+1)/2.

Author(s)

David Scott [e-mail], Christine Yang Dong [e-mail]

References

Scott, D. J., Würtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

Examples

  momRecursion(order = 12)

  #print out the matrix
  momRecursion(order = 12, "true")