R: Mode of a Stable Distribution

stable.mode {stable} | R Documentation

## 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, 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

    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")

[Package stable version 1.1.4 Index]