Stable innovations (S0 parametrization)

Standardized stable distribution \(S(\alpha, \beta, 1, 0; 0)\) in Nolan's S0 parametrization, computed with libstable4u. Parameters: stable_alpha (tail index, in (1, 2)) and stable_beta (skewness, in (-1, 1)).

Usage

gs_stable(
  lower = c(stable_alpha = 1.01, stable_beta = -0.99),
  upper = c(stable_alpha = 1.99, stable_beta = 0.99),
  start = c(stable_alpha = 1.8, stable_beta = 0)
)

Arguments

lower, upper, start

Optional named vectors replacing the default bounds and start value.

Value

A gs_dist object.

Details

The variance is infinite, so \(\sigma_t\) is a conditional scale, not a standard deviation. In GARCH mode the power is \(\delta = 1\) (absolute value GARCH), since \(E|z|^\delta\) is finite only for \(\delta < \alpha\). The mean is \(-\beta \tan(\pi\alpha/2)\).

References

Nolan, J. P. (2020). Univariate Stable Distributions. Springer.

Royuela-del-Val, J., Simmross-Wattenberg, F. and Alberola-Lopez, C. (2017). libstable: Fast, Parallel, and High-Precision Computation of alpha-Stable Distributions in R, C/C++, and MATLAB. Journal of Statistical Software, 78(1), 1-25.

Examples

d <- gs_stable()
d$log_density(c(-1, 0, 1), c(stable_alpha = 1.7, stable_beta = 0.2))
#> [1] -1.552620 -1.259304 -1.560797