Description Usage Arguments Details Functions Author(s) References See Also Examples
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
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14  dstable(x, loc = 0, disp = 1/sqrt(2), skew = 0, tail = 2,
npt = 501, up = 10, eps = 1e06, integration = "Romberg")
pstable(q, loc = 0, disp = 1/sqrt(2), skew = 0, tail = 2,
eps = 1e06)
qstable(p, loc = 0, disp = 1/sqrt(2), skew = 0, tail = 2,
eps = 1e06)
rstable(n = 1, loc = 0, disp = 1/sqrt(2), skew = 0, tail = 2,
eps = 1e06)
hstable(x, loc = 0, disp = 1/sqrt(2), skew = 0, tail = 2,
eps = 1e06)

x, q 
vector of quantiles. 
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. 
npt, up, integration 
As detailed herein – only available when using 
eps 
scalar giving the required precision in computation. 
p 
vector of probabilites. 
n 
number of observations. 
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 rightskewed, symmetric and leftskewed
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) logt
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
: density
pstable
: cdf
qstable
: quantiles
rstable
: random deviates
hstable
: hazard
Philippe Lambert (Catholic University of Louvain, Belgium, phlambert@stat.ucl.ac.be)
Jim Lindsey
Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns using regression models based on nonsymmetric stable distributions. Applied Statistics, 48, 409424.
stablereg
to fit generalized nonlinear regression models
for the stable distribution parameters.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  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))
stabledist::dstable(x=1, 1, 0)

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