BMTchangepars function

The BMT Distribution Parameter Conversion.

Parameter conversion for different parameterizations for the BMT distribution, with p3 and p4 tails weights ($\kappa_l$

and $\kappa_r$) or asymmetry-steepness parameters ($\zeta$ and $\xi$) and p1 and p2 domain (minimum and maximum) or location-scale (mean and standard deviation) parameters.

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

Arguments

• p3, p4: tails weights ($\kappa_l$ and $\kappa_r$) or asymmetry-steepness ($\zeta$ and $\xi$) parameters of the BMT distribution.
• type.p.3.4: type of parametrization associated 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 distribution.
• type.p.1.2: type of parametrization associated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.

Returns

BMTchangepars reparametrize 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.

Details

The BMT coefficient of asymmetry $-1 < \zeta < 1$ is

The BMT coefficient of steepness $0 < \xi < 1$ is

for $|\kappa_r - \kappa_l| < 1$.

The BMT distribution has mean c("$( d - c ) BMTmean(\\kappa_l, \\kappa_r) + \n$", "$c$") and standard deviation $( d - c ) BMTsd(\kappa_l, \kappa_r)$

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

Examples

# 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

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.

See Also

BMT for the BMT density, distribution, quantile function and random deviates.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co

and Alvaro Mauricio Montenegro Diaz [ths]