StableDistribution {fBasics}  R Documentation 
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. Two different cases are considered,
the first for the symmetric and the second for the
skewed distribution.
The functions are:
[dpqr]symstb  The symmetric stable distribution, 
[dpqr]stable  the skewed stable distribution, 
symstbSlider  interactive symmetric distribution display, 
stableSlider  interactive stable distribution display. 
dsymstb(x, alpha) psymstb(q, alpha) qsymstb(p, alpha) rsymstb(n, alpha) stableMode(alpha, beta) 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)) symstbSlider() stableSlider()
alpha, beta, gamma, delta 
value of the index parameter alpha with alpha = (0,2] ;
skewness parameter beta , in the range [1, 1];
scale parameter gamma ; and
shift parameter delta .

n 
number of observations, an integer value. 
p 
a numeric vector of probabilities. 
pm 
parameterization, an integer value by default pm=0 ,
the 'S0' parameterization.

x, q 
a numeric vector of quantiles. 
Symmetric Stable Distribution:
For the density and probability the approach of McCulloch is
implemented. Note, that McCulloch's approach has a density
precision of 0.000066 and a distribution precision of 0.000022
for alpha
in the range [0.84, 2.00].
Quantiles are evaluated from a root finding process via the
probability function. Thus, this leads to nonnegligible
errors for small quantiles, since the quantile evaluation
depends on the quality of the probability function.To achieve
higher precisions use the function stable
with argument
beta=0
.
For generation of random deviates the results of Chambers,
Mallows, and Stuck are used.
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 NonGaussian
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].
All values for the *symstb
and *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 functions symstbSlider
and stableSlider
display for educational purposes the densities and probabilities
of the symmetric and skew stable distributions.
McCulloch for the 'symstb' Fortran program, and
Diethelm Wuertz for the Rmetrics Rport.
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 NonGaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York, 632 pages.
Weron, A., Weron R. (1999); Computer Simulation of Levy alphaStable Variables and Processes, Preprint Technical Univeristy of Wroclaw, 13 pages.
## SOURCE("fBasics.13AStableDistribution") ## rsymstb  xmpBasics("\nStart: Symmetric Stable Distribuion: > ") par(mfcol = c(3, 2), cex = 0.7) set.seed(1953) r = rsymstb(n = 1000, alpha = 1.9) plot(r, type = "l", main = "symstb: alpha = 1.9") # Plot empirical density and compare with true density: hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue4") x = seq(5, 5, 0.1) lines(x, dsymstb(x = x, alpha = 1.9)) # Plot df and compare with true df: plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue4") lines(x, psymstb(x, alpha = 1.9)) # Compute quantiles: qsymstb(psymstb(q = seq(10, 10, 1), alpha = 1.9), alpha = 1.9) ## stable  xmpBasics("\nNext: Skew Stable Distribuion: > ") # Compared to R, this might be quite slow under SPlus ... 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") # Plot empirical density and compare with true density: hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue4") x = seq(5, 5, 0.4) lines(x, dstable(x = x, alpha = 1.9, beta = 0.3)) # Plot df and compare with true df: plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue4") lines(x, pstable(q = x, alpha = 1.9, beta = 0.3)) # Compute quantiles: qstable(pstable(seq(4, 4, 1), alpha = 1.9, beta = 0.3), alpha = 1.9, beta = 0.3) ## stable  xmpBasics("\nNext: Paramterization S1: > ") set.seed(1953) r = rstable(n = 1000, alpha = 1.9, beta = 0.3, pm = 1) plot(r, type = "l", main = "S1 stable: alpha=1.9 beta=0.3") # Plot empirical density and compare with true density: hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue4") x = seq(5, 5, 0.4) lines(x, dstable(x = x, alpha = 1.9, beta = 0.3)) # Plot df and compare with true df: plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue4") lines(x, pstable(q = x, alpha = 1.9, beta = 0.3, pm = 1)) # Compute quantiles: qstable(pstable(seq(4, 4, 1), alpha = 1.9, beta = 0.3, pm = 1), alpha = 1.9, beta = 0.3, pm = 1)