gat-moment: Moment Functions of Asymmetric Student-t distribution

In dan9401/st: Asymmetric Student t-distributions

Description

The mean, standard deviation, skewness, kurtosis functions, as well as the raw and central moments of GAT distribution

Usage

gatMean(mu = 0, sigma = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf,
  pars = NULL, method = c("analytical", "numerical"))

gatMoments(mu = 0, sigma = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf,
  pars = NULL, method = c("analytical", "numerical"),
  type = c("excess", "regular"))

gatRawMoment(n, mu = 0, phi = 1, alpha = 0.5, r = 2, c = 2,
  nu = Inf, pars = NULL, method = c("analytical", "numerical"))

gatCentralMoment(n, mu = 0, phi = 1, alpha = 0.5, r = 2, c = 2,
  nu = Inf, pars = NULL, method = c("analytical", "numerical"))

Arguments

mu	location parameter
sigma	scale parameter, sigma > 0
alpha	skewness parameter, 0 < alpha < 1
nu1	degrees of freedom / tail parameter for the left tail, nu1 > 0
nu2	degrees of freedom / tail parameter for the right tail, nu2 > 0
pars	a vector that contains mu, phi, alpha, r, c, nu, if pars is specified, mu, phi, alpha, r, c, nu should not be specified
method	method used to calculate the moment(s), one of 'analytical' and 'numerical'
type	type of kurtosis calculated, one of 'excess' and 'regular'
n	order of (raw/central) moment to be calculated
moment	the moment to be calculated, one of 'mean', 'sd', 'skew', 'kurt'

Details

Function gatMoment calculates one of mean, standard deviation, skewness and kurtosis of the distribution, while gatMoment calculates all 4 of them.
Function gatRawMoment returns E[X^n], while function gatCentralMoment returns E[(X-μ)^n]

The moments for GAT follow the general rule of student t distribution,

• mean is only defined for nu > 1,

• variance/standard deviation is finite when nu > 2, infinite when 1 < nu < 2, otherwise undefined,

• skewness is defined when nu > 3,

• kurtosis is finite when nu > 4, infinite when 2 < nu <= 4, otherwise undefined.

References

Zhu, D., & Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics, 157(2), 297-305.https://www.sciencedirect.com/science/article/pii/S0304407610000266 https://econpapers.repec.org/paper/circirwor/2009s-13.htm

Examples

# The parameter values are specially set for a volatile portfolio.
pars <- c(0.12, 0.6, 0.6, 6, 5)
gatMoment("sd", pars = pars, method = "numerical")
gatMoments(pars = pars)

dan9401/st documentation built on Sept. 5, 2020, 5:16 a.m.