rstable1: Random numbers from (Skew) Stable Distributions

Description

Generate random numbers of the stable distribution

S (α, β, γ, δ; k)

with characteristic exponent $\alpha\in(0,2]$, skewness $\beta\in[-1,1]$, scale $\gamma\in[0,\infty)$, and location $\delta\in\mathbf{R}$; see Nolan (2010) for the parameterization $k\in{0,1}$. The case $\gamma=0$ is understood as the unit jump at $\delta$.

Usage

rstable1(n, alpha, beta, gamma = 1, delta = 0, pm = 1)

Arguments

an integer, the number of observations to generate.

alpha

characteristic exponent $\alpha\in(0,2]$.

beta

skewness $\beta\in[-1,1]$.

gamma

scale $\gamma\in[0,\infty)$.

delta

location $\delta\in\mathbf{R}$.

0 or 1, denoting which parametrization (as by Nolan) is used.

Value

A numeric vector of length n containing the generated random variates.

Details

We use the approach of John Nolan for generating random variates of stable distributions. The function rstable1 provides two basic parametrizations, by default,

pm = 1, the so called S, S1, or 1 parameterization. This is the parameterization used by Samorodnitsky and Taqqu (1994), and is a slight modification of Zolotarev's (A) parameterization. It is the form with the most simple form of the characteristic function; see Nolan (2010, p. 8).

pm = 0 is the S0 parameterization: based on the (M) representation of Zolotarev for an alpha stable distribution with skewness beta. Unlike the Zolotarev (M) parameterization, gamma and delta are straightforward scale and shift parameters. This representation is continuous in all 4 parameters.

References

Chambers, J. M., Mallows, C. L., and Stuck, B. W. (1976), A Method for Simulating Stable Random Variables, J. Amer. Statist. Assoc. 71, 340--344.

Nolan, J. P. (2012), Stable Distributions---Models for Heavy Tailed Data, Birkhaeuser, in progress.

Samoridnitsky, G. and Taqqu, M. S. (1994), Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York.

Examples

Run this code

# Generate and plot a series of stable random variates
   set.seed(1953)
   r <- rstable1(n = 1000, alpha = 1.9, beta = 0.3)
   plot(r, type = "l", main = "stable: alpha=1.9 beta=0.3",
        col = "steelblue"); grid()

   hist(r, "Scott", prob = TRUE, ylim = c(0,0.3),
        main = "Stable S(1.9, 0.3; 1)")
   lines(density(r), col="red2", lwd = 2)

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