ast-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 AST distribution

Usage

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

astMoment(
  moment = c("mean", "sd", "var", "skew", "kurt"),
  mu = 0,
  s = 1,
  alpha = 0.5,
  nu1 = Inf,
  nu2 = Inf,
  pars = NULL,
  method = c("analytical", "numerical"),
  type = c("excess", "regular")
)

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

astRawMoment(
  n,
  mu = 0,
  s = 1,
  alpha = 0.5,
  nu1 = Inf,
  nu2 = Inf,
  pars = NULL,
  method = c("analytical", "numerical")
)

astCentralMoment(
  n,
  mu,
  s,
  alpha,
  nu1,
  nu2,
  pars = NULL,
  method = c("analytical", "numerical")
)

Arguments

mu	location parameter
s	scale parameter, s > 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, s, alpha, nu1, nu2, if pars is specified, mu, s, alpha, nu1, nu2 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', 'var', 'skew', 'kurt'
type	type of kurtosis calculated, one of 'excess' and 'regular'
n	order of (raw/central) moment to be calculated

Details

Function astMoment calculates one of mean, standard deviation, skewness and kurtosis of the distribution, while astMoment calculates all 4 of them.
Function astRawMoment returns E[X^n], while function astCentralMoment returns E[(X-\mu)^n]

The moments for AST 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. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.1016/j.jeconom.2010.01.013")}

Examples

# The parameter values are specially set for a volatile portfolio.
pars <- c(0.12, 0.6, 0.6, 3, 5)

astMoment("sd", pars = pars, method = "numerical")
astMoments(pars = pars)

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