# unifed.kappa: Cumulant generator of the unifed distribution In unifed: The Unifed Distribution

## Description

Cumulant generator of the unifed distribution

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```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.

## Details

The cumulant generator of the unifed distribution is defined as

/ | / theta \ | |e - 1| kappa(theta) = | log |----------| if theta ! = 0 | \ theta / | \ 0 if theta = 0

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

## 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`.

## References

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

Jørgensen, Bent (1997). The Theory of Dispersion Models. Chapman & Hall, London.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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)) ```

unifed documentation built on April 14, 2021, 1:07 a.m.