stable.mode {stable}

Mode of a Stable Distribution

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

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, [e-mail]) 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

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")