Stable Distribution Function

Description

A collection and description of functions to compute density, distribution function, quantile function and to generate random variates, the stable distribution, and the stable mode.

The functions are:

Usage

dstable(x, alpha, beta, gamma = 1, delta = 0, pm = c(0, 1, 2))
pstable(q, alpha, beta, gamma = 1, delta = 0, pm = c(0, 1, 2))
qstable(p, alpha, beta, gamma = 1, delta = 0, pm = c(0, 1, 2))
rstable(n, alpha, beta, gamma = 1, delta = 0, pm = c(0, 1, 2))

stableMode(alpha, beta)

stableSlider()

Arguments

Details

Skew Stable Distribution:

The function uses the approach of J.P. Nolan for general stable distributions. Nolan derived expressions in form of integrals based on the charcteristic function for standardized stable random variables. These integrals are numerically evaluated using R's function integrate.
"S0" parameterization [pm=0]: 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, and gives an intuitive meaning to gamma and delta that is lacking in other parameterizations.
"S" or "S1" parameterization [pm=1]: the parameterization used by Samorodnitsky and Taqqu in the book Stable Non-Gaussian Random Processes. It is a slight modification of Zolotarev's (A) parameterization.
"S*" or "S2" parameterization [pm=2]: a modification of the S0 parameterization which is defined so that (i) the scale gamma agrees with the Gaussian scale (standard dev.) when alpha=2 and the Cauchy scale when alpha=1, (ii) the mode is exactly at delta.
"S3" parameterization [pm=3]: an internal parameterization. The scale is the same as the S2 parameterization, the shift is -beta*g(alpha), where g(alpha) is defined in Nolan [1999].

Value

All values for the *stable functions are numeric vectors: d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates.
The function stableMode returns a numeric value, the location of the stable mode.
The function stableSlider displays for educational purposes the densities and probabilities of the skew stable distribution.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

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. (1999); Stable Distributions, Preprint, University Washington DC, 30 pages.

Nolan J.P. (1999); Numerical Calculation of Stable Densities and Distribution Functions, Preprint, University Washington DC, 16 pages.

Samoridnitsky G., Taqqu M.S. (1994); Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York, 632 pages.

Weron, A., Weron R. (1999); Computer Simulation of Levy alpha-Stable Variables and Processes, Preprint Technical Univeristy of Wroclaw, 13 pages.

Examples

    
## stable - 
   # Plot rvs Series
   set.seed(1953)
   r = rstable(n = 1000, alpha = 1.9, beta = 0.3)
   plot(r, type = "l", main = "stable: alpha=1.9 beta=0.3", 
     col = "steelblue")
   grid()
 
## stable -  
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", 
     col = "steelblue")
   x = seq(-5, 5, 0.4)
   lines(x, dstable(x = x, alpha = 1.9, beta = 0.3))
   
## stable -   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", pch = 19, 
     col = "steelblue")
   lines(x, pstable(q = x, alpha = 1.9, beta = 0.3))
   grid()
   
## stable -
   # Compute quantiles:
   qstable(pstable(seq(-4, 4, 1), alpha = 1.9, beta = 0.3), 
     alpha = 1.9, beta = 0.3)