# The characteristic function of a stable distribution

### Description

It computes the theoretical characteristic function of a stable distribution for two different parametrizations. It is used in the vignette to illustrate the estimation of the parameters using GMM.

### Usage

1 | ```
charStable(theta, tau, pm = 0)
``` |

### Arguments

`theta` |
Vector of parameters of the stable distribution. See details. |

`tau` |
A vector of numbers at which the function is evaluated. |

`pm` |
The type of parametization. It takes the values 0 or 1. |

### Details

The function returns the vector *Ψ(θ,τ,pm)* defined as *E(e^{ixτ}*, where *τ* is a vector of real numbers, *i* is the imaginary number, *x* is a stable random variable with parameters *θ* = *(α,β,γ,δ)* and `pm`

is the type of parametrization. The vector of parameters are the characteristic exponent, the skewness, the scale and the location parameters, respectively. The restrictions on the parameters are: *α \in (0,2]*, *β\in [-1,1]* and *γ>0*. For mode details see Nolan(2009).

### Value

It returns a vector of complex numbers with the dimension equals to `length(tau)`

.

### References

Nolan J. P. (2009), Stable Disttributions.
*Math/Stat Department, American University*.
URL http://academic2.american.edu/~jpnolan/stable/stable.html.

### Examples

1 2 3 4 5 6 | ```
# GMM is like GLS for linear models without endogeneity problems
pm <- 0
theta <- c(1.5,.5,1,0)
tau <- seq(-3, 3, length.out = 20)
char_fct <- charStable(theta, tau, pm)
``` |