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

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).

1 | ```
stable.mode(loc, disp, skew, tail)
``` |

`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. |

`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)*

.

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

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.

`stable`

for more details on the stable
distribution.

`stablereg`

to fit generalized linear models for the
stable distribution parameters.

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