fit_mvst {fitHeavyTail}

Estimate parameters of a multivariate (generalized hyperbolic) skewed t distribution to fit data

Description

Estimate parameters of a multivariate (generalized hyperbolic) skewed Student's t distribution to fit data, namely, the location vector, the scatter matrix, the skewness vector, and the degrees of freedom. The estimation is based on the maximum likelihood estimation (MLE) and the algorithm is obtained from the expectation-maximization (EM) method.

Usage

fit_mvst(
  X,
  nu = NULL,
  gamma = NULL,
  initial = NULL,
  max_iter = 500,
  ptol = 0.001,
  ftol = Inf,
  PXEM = TRUE,
  return_iterates = FALSE,
  verbose = FALSE
)

Arguments

X

Data matrix containing the multivariate time series (each column is one time series).

nu

Degrees of freedom of the skewed t distribution (otherwise it will be iteratively estimated).

gamma

Skewness vector of the skewed t distribution (otherwise it will be iteratively estimated).

initial

List of initial values of the parameters for the iterative estimation method. Possible elements include:

• nu: default is 4,

• mu: default is the data sample mean,

• gamma: default is the sample skewness vector,

• scatter: default follows from the scaled sample covariance matrix,

max_iter

Integer indicating the maximum number of iterations for the iterative estimation method (default is 500).

ptol

Positive number indicating the relative tolerance for the change of the variables to determine convergence of the iterative method (default is 1e-3).

ftol

Positive number indicating the relative tolerance for the change of the log-likelihood value to determine convergence of the iterative method (default is Inf, so it is not active). Note that using this argument might have a computational cost as a convergence criterion due to the computation of the log-likelihood (especially when X is high-dimensional).

PXEM

Logical value indicating whether to use the parameter expansion (PX) EM method to accelerating the convergence.

return_iterates

Logical value indicating whether to record the values of the parameters (and possibly the log-likelihood if ftol < Inf) at each iteration (default is FALSE).

verbose

Logical value indicating whether to allow the function to print messages (default is FALSE).

Details

This function estimates the parameters of a (generalized hyperbolic) multivariate Student's t distribution (mu, scatter, gamma and nu) to fit the data via the expectation-maximization (EM) algorithm.

Value

A list containing (possibly) the following elements:

mu

Location vector estimate (not the mean).

gamma

Skewness vector estimate.

scatter

Scatter matrix estimate.

nu

Degrees of freedom estimate.

mean

Mean vector estimate:

  mean = mu + nu/(nu-2) * gamma

cov

Covariance matrix estimate:

  cov = nu/(nu-2) * scatter + 2*nu^2 / (nu-2)^2 / (nu-4) * gamma*gamma'

converged

Boolean denoting whether the algorithm has converged (TRUE) or the maximum number of iterations max_iter has been reached (FALSE).

num_iterations

Number of iterations executed.

cpu_time

Elapsed overall CPU time.

log_likelihood_vs_iterations

Value of log-likelihood over the iterations (if ftol < Inf).

iterates_record

Iterates of the parameters (mu, scatter, nu, and possibly log_likelihood (if ftol < Inf)) along the iterations (if return_iterates = TRUE).

cpu_time_at_iter

Elapsed CPU time at each iteration (if return_iterates = TRUE).

Author(s)

Rui Zhou, Xiwen Wang, and Daniel P. Palomar

References

Aas Kjersti and Ingrid Hobæk Haff. "The generalized hyperbolic skew Student’s t-distribution," Journal of financial econometrics, pp. 275-309, 2006.

See Also

fit_mvt

Examples

library(mvtnorm)       # to generate heavy-tailed data
library(fitHeavyTail)

# parameter setting
N <- 5
T <- 200
nu <- 6
mu <- rnorm(N)
scatter <- diag(N)
gamma <- rnorm(N)   # skewness vector

# generate GH Skew t data
taus <- rgamma(n = T, shape = nu/2, rate = nu/2)
X <- matrix(data = mu, nrow = T, ncol = N, byrow = TRUE) +
     matrix(data = gamma, nrow = T, ncol = N, byrow = TRUE) / taus +
     rmvnorm(n = T, mean = rep(0, N), sigma = scatter) / sqrt(taus)

# fit skew t model
fit_mvst(X)

# setting lower limit for nu (e.g., to guarantee existence of co-skewness and co-kurtosis matrices)
options(nu_min = 8.01)
fit_mvst(X)