| stable {stable} | R Documentation |
These functions provide information about the stable distribution
with the location, the dispersion, the skewness and the tail thickness
respectively modelled by the parameters loc, disp,
skew and tail.
dstable, pstable, qstable and hstable
compute the density, the distribution, the quantile and the hazard functions
of a stable variate. rstable generates random deviates with
the prescribed stable distribution.
loc is a location parameter in the same way as the mean
in the normal distribution: it can take any real value.
disp is a dispersion parameter in the same way as the standard
deviation in the normal distribution: it can take any positive value.
skew is a skewness parameter: it can take any value in (-1,1).
The distribution is right-skewed, symmetric and left-skewed
when skew is negative, null or positive respectively.
tail is a tail parameter (often named the characteristic exponent):
it can take any value in (0,2) (with tail=1 and tail=2
yielding the Cauchy and the normal distributions respectively
when symmetry holds).
If loc, disp, skew, or tail are not
specified they assume the default values of 0, 1/sqrt(2),
0 and 2 respectively. This corresponds to a normal
variate with mean=0 and variance=1/2 disp^2.
The stable characteristic function is given by
phi(t) = i loc t - disp |t|^tail [1+i skew sign(t) omega(t,tail)]
where
omega(t,tail) = (2/pi) log|t|
when tail=1, and
omega(t,tail) = tan(pi alpha / 2)
otherwise.
The characteristic function is inverted using Fourier's transform
to obtain the corresponding stable density. This inversion requires the
numerical evaluation of an integral from 0 to infinity.
Two algorithms
are proposed for this. The default is Romberg's method
(integration="Romberg") which is used to evaluate the integral
with an error bounded by eps.
The alternative method is Simpson's integration
(integration="Simpson"): it approximates the
integral from 0 to infinity by an integral
from 0 to up
with npt points subdividing (O, up).
These three extra arguments – integration, up and
npt – are only available when using dstable.
The other functions are all based on Romberg's algorithm.
dstable(y, loc=0, disp=1/sqrt(2), skew=0, tail=2,
npt=501, up=10, eps=1.0e-6, integration="Romberg")
pstable(y, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
qstable(q, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
hstable(y, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
rstable(n=1, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
y,q |
vector of quantiles. |
p |
vector of probabilites. |
n |
number of observations. |
loc |
vector of (real) location parameters. |
disp |
vector of (positive) dispersion parameters. |
skew |
vector of skewness parameters (in [-1,1]). |
tail |
vector of parameters (in [0,2]) related to the tail thickness. |
eps |
scalar giving the required precision in computation. |
Philippe Lambert (Catholic University of Louvain, Belgium, phlambert@stat.ucl.ac.be) and Jim Lindsey.
Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns using regression models based on non-symmetric stable distributions. Applied Statistics, 48, 409-424.
stablereg to fit generalized nonlinear regression models
for the stable distribution parameters.
stable.mode to compute the mode of a stable
distribution.
par(mfrow=c(2,2)) x <- seq(-5,5,by=0.1) # Influence of loc (location) plot(x,dstable(x,loc=-2,disp=1/sqrt(2),skew=-0.8,tail=1.5), type="l",ylab="",main="Varying LOCation") lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=-0.8,tail=1.5)) lines(x,dstable(x,loc=2,disp=1/sqrt(2),skew=-0.8,tail=1.5)) # Influence of disp (dispersion) plot(x,dstable(x,loc=0,disp=0.5,skew=0,tail=1.5), type="l",ylab="",main="Varying DISPersion") lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5)) lines(x,dstable(x,loc=0,disp=0.9,skew=0,tail=1.5)) # Influence of skew (skewness) plot(x,dstable(x,loc=0,disp=1/sqrt(2),skew=-0.8,tail=1.5), type="l",ylab="",main="Varying SKEWness") lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5)) lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0.8,tail=1.5)) # Influence of tail (tail) plot(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=0.8), type="l",ylab="",main="Varying TAIL thickness") lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5)) lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=2))