Stable Family of Distributions

DESCRIPTION:

Returns random deviates from the stable family of probability distributions.

USAGE:

rstab(n, index, skewness=0, bigdata=F) 

REQUIRED ARGUMENTS:

n
sample size. If length(n) is larger than 1, then length(n) random values are returned.
index
number from the interval (0, 2]. An index of 2 corresponds to the normal, 1 to the Cauchy. Smaller values mean longer tails. This may be a vector or bdVector. Missing values ( NAs) are allowed.

OPTIONAL ARGUMENTS:

skewness
number giving the modified skewness (see Chambers et al., 1976). This is not the ordinary skewness of a distribution (see ). Positive values correspond to a long right tail, where the mean (if it exists) is greater than the median. Negative values correspond to a long left tail. The absolute value of skewness should not exceed 1.
bigdata
a logical value; if TRUE, an object of type bdVector is returned. Otherwise, a vector object is returned. This argument can be used only if the bigdata library section has been loaded.

VALUE:

random sample from the specified stable distribution. If index or skewness are shorter than the sample size, their values are re-used cyclically.

DETAILS:

A nonlinear transformation of 2 independent uniforms is performed.

Note that there are no density, probability or quantile functions supplied for this distribution. The efficient computation of such values is an open problem.

BACKGROUND:

Stable distributions are of considerable mathematical interest. Each stable distribution is the limit distribution of a suitably scaled sum of independent and identically distributed random variables. Statistically, they are used mostly when an example of a very long-tailed distribution is required. For small values of index, the distribution degenerates to point mass at 0.

For details on the uniform random number generator implemented in S-PLUS, see the set.seed help file.

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.

Loéve, M. (1977). Probability Theory I. (fourth edition) Springer-Verlag, New York.

SEE ALSO:

, , .

EXAMPLES:

set.seed(0)
hist(rstab(200, 1.5, .5)) #fairly long tails, positive skewness
# Note -- if you repeat this many times without setting the seed, 
# many of the plots will show negative skewness.  These stable
# distributions may have outliers on either side.