rstable {CircStats}

Random Generator for Stable Family of Distributions

Description

Returns random deviates from the stable family of probability distributions.

Usage

rstable(n, scale = 1, index = stop("no index arg"), skewness = 0)

Arguments

n

sample size

index

number from the interval (0, 2]. An index of 2 corresponds to the normal, 1 to the Cauchy. Smaller values mean longer tails.

skewness

number giving the modified skewness (see Chambers et al., 1976). Negative values correspond to skewness to the left (the median is smaller than the mean, if it exists), and positive values correspond to skewness to the right (the median is larger than the mean). The absolute value of skewness should not exceed 1.

scale

the scale of the distribution.

Details

This function returns a random variate from the Levy skew stable distribution with index=\alpha, scale=c and skewness=\beta. The skewness parameter must lie in the range [-1,1] while the index parameter must lie in the range (0,2]. The Levy skew stable probability distribution is defined by a fourier transform,

p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha (1-i \beta sign(t) \tan(\pi\alpha/2)))

When \alpha=1 the term \tan(\pi \alpha/2) is replaced by -(2/\pi)\log|t|. For \alpha=2 the distribution reduces to a Gaussian distribution with \sigma = \sqrt{2} scale and the skewness parameter has no effect. For \alpha < 1 the tails of the distribution become extremely wide. The symmetric distribution corresponds to \beta=0.

The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \alpha, \beta) then the sum $Y = X_1 + X_2 + ... + X_N$ will also be distributed as an alpha-stable variate, p(N^{1/\alpha} c, \alpha, \beta).

There is no explicit solution for the form of p(x) and there are no density, probability or quantile functions supplied for this distribution.

Value

random sample from the specified stable distribution.

References

Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association 71, 340-344.

Lo\"gae\"ve, M. (1977). Probability Theory I. (fourth edition) Springer-Verlag, New York.

See Also

rnorm, rcauchy.

Examples

hist(rstable(200, 1.5, .5)) #fairly long tails, skewed right