stable.mode: Mode of a Stable Distribution

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/stable.r

Description

This function gives a reliable approximation to the mode of a stable distribution with location, dispersion, skewness and tail thickness specified by the parameters loc, disp, skew and tail. tail must be in (1,2).

Usage

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stable.mode(loc, disp, skew, tail)

Arguments

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 [1,2]) related to the tail thickness.

Details

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).

The simplest empirical formula found to give a satisfactory approximation to the mode for values of tail in (1,2) is

loc+disp*a*skew*exp(-b*abs(skew))

with

a = 1.7665114+1.8417675*tail-2.2954390*tail^2+0.4666749*tail^3

and

b = -0.003142967+632.4715*tail*exp(-7.106035*tail)

.

Value

A list of size 3 giving the mode, a and b.

Author(s)

Philippe Lambert (Catholic University of Louvain, Belgium, phlambert@stat.ucl.ac.be) and Jim Lindsey.

References

Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns using regression models based on non-symmetric stable distributions. Applied Statistics, 48, 409-424.

See Also

stable for more details on the stable distribution.

stablereg to fit generalized linear models for the stable distribution parameters.

Examples

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x <- seq(-5,5,by=0.1)
plot(x,dstable(x,loc=0,disp=1,skew=-1,tail=1.5),type="l",ylab="f(x)")
xhat <- stable.mode(loc=0,disp=1,skew=-1,tail=1.5)$ytilde
fxhat <- dstable(xhat,loc=0,disp=1,skew=-1,tail=1.5)
lines(c(xhat,xhat),c(0,fxhat),lty="dotted")

stable documentation built on May 2, 2019, 6:39 a.m.