# 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. ```r 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 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. ## Returns `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`. ## Details The BMT coefficient of asymmetry $-1 < \zeta < 1$ is $$ \kappa_r - \kappa_l $$ The BMT coefficient of steepness $0 < \xi < 1$ is $$ (\kappa_r +\kappa_l - |\kappa_r - \kappa_l|) / (2 (1 - |\kappa_r - \kappa_l|)) $$ 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 ```r # 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]

BMTchangepars function