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

In dan9401/skewtDist: Asymmetric Student t-Distributions

Moment Functions of Asymmetric Student-t distribution

Description

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

Usage

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

gatMoment(
  moment = c("mean", "sd", "var", "skew", "kurt"),
  mu = 0,
  phi = 1,
  alpha = 0.5,
  r = 2,
  c = 2,
  nu = Inf,
  pars = NULL,
  method = c("analytical", "numerical"),
  type = c("excess", "regular")
)

gatMoments(
  mu = 0,
  phi = 1,
  alpha = 0.5,
  r = 2,
  c = 2,
  nu = 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
phi	scale parameter, phi > 0
alpha	skewness parameter, 0 < alpha < 1
r	tail power asymmetry parameter r > 0
c	scale asymmetry parameter r > 0
nu	degrees of freedom / tail parameter
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'
moment	the moment to be calculated, one of 'mean', 'sd', 'skew', 'kurt'
type	type of kurtosis calculated, one of 'excess' and 'regular'
n	order of (raw/central) moment to be calculated

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-\mu)^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

Baker, R. D. (2016). A new asymmetric generalisation of the t-distribution. arXiv preprint arXiv:1606.05203. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.48550/arXiv.1606.05203")}

Examples

# The parameter values are specially set for a volatile portfolio.
pars <- c(0.12, 0.6, 1.5, 1.2, 2, 5)

gatMoment("sd", pars = pars, method = "numerical")
gatMoments(pars = pars)

dan9401/skewtDist documentation built on Jan. 6, 2025, 9:14 a.m.