unifed (version 1.1.6)

unifed.kappa: Cumulant generator of the unifed distribution

Description

Cumulant generator of the unifed distribution

Usage

unifed.kappa(theta)
unifed.kappa.prime(theta)
unifed.kappa.double.prime(theta)
unifed.kappa.prime.inverse(mu, ...)
unifed.kappa.prime.inverse.one(mu, tol = 1e-07, maxit = 1e+07)

Arguments

theta

A numeric vector.

mu

A vector of numbers between 0 and 1

...

Other parameters of unifed.kappa.prime.inverse.one

tol

Tolerance level. The algorithm stops if the proportional difference between the new and old value of an iteration is less or equal than this number.

maxit

Maximum number of iterations of the algorithm to look for convergence.

Value

unifed.kappa returns a vector that contains the cumulant generator of the unifed distribution applied to each element of theta.

unifed.kappa.prime returns a vector that contains the derivative of the cumulant generator of the unifed distribution for each element of theta.

unifed.kappa.double.prime returns a vector that contains the second derivative of the cumulant generator of the unifed distribution for each element of theta.

unifed.kappa.prime.inverse returns a vector with unifed.kappa.prime.inverse.one evaluated at every entry of mu.

unifed.kappa.prime.inverse.one if the tolerance level is reached within maxit iterations, the function returns the value of the last iteration. Otherwise it returns NA.

Details

The cumulant generator of the unifed distribution is defined as

$$ \kappa(\theta)=\left\{ \begin{array}{ll} \log\left(\frac{e^\theta-1}{\theta}\right)& if \theta \neq 0\\ 0 & \mbox{if }\theta=0 \end{array} \right..$$

unifed.kappa.prime.inverse.one uses the Newthon-Raphson method for finding the inverse of unifed.kappa.prime for a single value.

References

Quijano Xacur, O.A. The unifed distribution. J Stat Distrib App 6, 13 (2019). doi:10.1186/s40488-019-0102-6.

J<U+00F8>rgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall, London.

Examples

# NOT RUN {
unifed.kappa(1)
unifed.kappa(-5:5)


unifed.kappa.prime(4.5)


unifed.kappa.double.prime(0)


unifed.kappa.prime.inverse(0.5)
unifed.kappa.prime.inverse(c(0.1,0.7,0.9))


# }