# BMTchangepars: The BMT Distribution Parameter Conversion. In BMT: The BMT Distribution

## Description

Parameter conversion for different parameterizations for the BMT distribution, with `p3` and `p4` tails weights (κ_l and κ_r) or asymmetry-steepness parameters (ζ and ξ) and `p1` and `p2` domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

## Usage

 ```1 2``` ```BMTchangepars(p3, p4, type.p.3.4 = "t w", p1 = NULL, p2 = NULL, type.p.1.2 = NULL) ```

## Arguments

 `p3, p4` tails weights (κ_l and κ_r) or asymmetry-steepness (ζ and ξ) parameters of the BMT distribution. `type.p.3.4` type of parametrization asociated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization. `p1, p2` domain (minimum and maximum) or location-scale (mean and standard deviation) parameters of the BMT ditribution. `type.p.1.2` type of parametrization asociated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

## Details

The BMT coefficient of asymmetry -1 < ζ < 1 is

κ_r - κ_l

The BMT coefficient of steepness 0 < ξ < 1 is

(κ_r + κ_l - |κ_r - κ_l|) / (2 (1 - |κ_r - κ_l|))

for |κ_r - κ_l| < 1.

The BMT distribution has mean ( d - c ) BMTmean(κ_l, κ_r) + c and standard deviation ( d - c ) BMTsd(κ_l, κ_r)

From these equations, we can go back and forth with each parameterization.

## Value

`BMTchangepars` reparametrizes `p3`, `p4`, `p1`, `p2` according to the alternative parameterizations from the given `type.p.3.4` and `type.p.1.2`. `BMTchangepars` returns a list with the alternative arguments to those received.

The arguments are recycled to the length of the result. Only the first elements of `type.p.3.4` and `type.p.1.2` are used.

If `type.p.3.4 == "t w"`, `p3 < 0` and `p3 > 1` are errors and return `NaN`.

If `type.p.3.4 == "a-s"`, `p3 < -1` and `p3 > 1` are errors and return `NaN`.

`p4 < 0` and `p4 > 1` are errors and return `NaN`.

If `type.p.1.2 == "c-d"`, `p1 >= p2` is an error and returns `NaN`.

If `type.p.1.2 == "l-s"`, `p2 <= 0` is an error and returns `NaN`.

## Author(s)

Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co and Alvaro Mauricio Montenegro Diaz [ths]

## References

Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.

`BMT` for the BMT density, distribution, quantile function and random deviates.
 ``` 1 2 3 4 5 6 7 8 9 10``` ```# BMT on [0,1] with left tail weight equal to 0.25 and # right tail weight equal to 0.75 parameters <- BMTchangepars(0.25, 0.75, "t w") parameters # Parameters of the BMT in the asymmetry-steepness parametrization # BMT with mean equal to 0, standard deviation equal to 1, # asymmetry coefficient equal to 0.5 and # steepness coefficient equal to 0.75 parameters <- BMTchangepars(0.5, 0.5, "a-s", 0, 1, "l-s") parameters # Parameters of the BMT in the tail weight and domain parametrization ```