# The BMT Distribution Moments, Moment-Generating Function and Characteristic Function. Any raw, central or standardized moment, the moment-generating function and the characteristic function 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 BMTmoment( p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d", order, type = "standardised", method = "quadrature" ) BMTmgf(s, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d") BMTchf(s, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d") mBMT(order, p3, p4, type.p.3.4, p1, p2, type.p.1.2) ``` ## 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. - `order`: order of the moment. - `type`: type of the moment: raw, central or standardized (default). - `method`: method to obtain the moment: exact formula or Chebyshev-Gauss quadrature (default). - `s`: variable for the moment-generating and characteristic functions. ## Returns `BMTmoment` gives any raw, central or standardized moment, `BMTmgf` the moment-generating function and `BMTchf` the characteristic function The arguments are recycled to the length of the result. Only the first elements of `type.p.3.4`, `type.p.1.2`, `type` and `method` 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 See References. ## Examples ```r layout(matrix(1:4, 2, 2, TRUE)) s <- seq(-1, 1, length.out = 100) # BMT on [0,1] with left tail weight equal to 0.25 and # right tail weight equal to 0.75 BMTmoment(0.25, 0.75, order = 5) # hyperskewness by Gauss-Legendre quadrature BMTmoment(0.25, 0.75, order = 5, method = "exact") # hyperskewness by exact formula mgf <- BMTmgf(s, 0.25, 0.75) # moment-generation function plot(s, mgf, type="l") chf <- BMTchf(s, 0.25, 0.75) # characteristic function # BMT on [0,1] with asymmetry coefficient equal to 0.5 and # steepness coefficient equal to 0.5 BMTmoment(0.5, 0.5, "a-s", order = 5) BMTmoment(0.5, 0.5, "a-s", order = 5, method = "exact") mgf <- BMTmgf(s, 0.5, 0.5, "a-s") plot(s, mgf, type="l") chf <- BMTchf(s, 0.5, 0.5, "a-s") # BMT on [-1.783489, 3.312195] with # left tail weight equal to 0.25 and # right tail weight equal to 0.75 BMTmoment(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d", order = 5) BMTmoment(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d", order = 5, method = "exact") mgf <- BMTmgf(s, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") plot(s, mgf, type="l") chf <- BMTchf(s, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") # BMT with mean equal to 0, standard deviation equal to 1, # asymmetry coefficient equal to 0.5 and # steepness coefficient equal to 0.5 BMTmoment(0.5, 0.5, "a-s", 0, 1, "l-s", order = 5) BMTmoment(0.5, 0.5, "a-s", 0, 1, "l-s", order = 5, method = "exact") mgf <- BMTmgf(s, 0.5, 0.5, "a-s", 0, 1, "l-s") plot(s, mgf, type="l") chf <- BMTchf(s, 0.5, 0.5, "a-s", 0, 1, "l-s") ``` ## References Torres-Jimenez, C. J. and Montenegro-Diaz, A. M. (2017, September), **An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution**. ArXiv e-prints. [https://arxiv.org/abs/1709.05534](https://arxiv.org/abs/1709.05534). 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 `BMTcentral`, `BMTdispersion`, `BMTskewness`, `BMTkurtosis` for specific descriptive measures or moments. ## Author(s) Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co

BMTmoments function