# BMT: The BMT Distribution. In BMT: The BMT Distribution

## Description

Density, distribution, quantile function, random number generation 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 3 4 5 6 7 8 9 10 dBMT(x, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d", log = FALSE) pBMT(q, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d", lower.tail = TRUE, log.p = FALSE) qBMT(p, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d", lower.tail = TRUE, log.p = FALSE) rBMT(n, p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d") 

## Arguments

 x, q vector of quantiles. 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. log, log.p logical; if TRUE, probabilities p are given as log(p). lower.tail logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. p vector of probabilities. n number of observations. If length(n) > 1, the lenght is taken to be the number required

## Details

The BMT distribution with tails weights and domain parametrization (type.p.3.4 = "t w" and type.p.1.2 = "c-d") has quantile function

(d - c) [3 t_p ( 1 - t_p )^2 κ_l - 3 t_p^2 ( 1 - t_p ) κ_r + t_p^2 ( 3 - 2 t_p ) ] + c

where 0 ≤ p ≤ 1, t_p = 1/2 - \cos ( [\arccos ( 2 p - 1 ) - 2 π] / 3 ), and 0 < κ_l < 1 and 0 < κ_r < 1 are, respectively, related to left and right tail weights or curvatures.

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.

## Value

dBMT gives the density, pBMT the distribution function, qBMT the quantile function, and rBMT generates random deviates.

The length of the result is determined by n for rBMT, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments 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. and Montenegro-Diaz, A. M. (2017, September), An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution. ArXiv e-prints.

Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.

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.

BMTcentral, BMTdispersion, BMTskewness, BMTkurtosis, BMTmoments for descriptive measures or moments. BMTchangepars for parameter conversion between different parametrizations.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 # BMT on [0,1] with left tail weight equal to 0.25 and # right tail weight equal to 0.75 z <- seq(0, 1, length.out = 100) F1 <- pBMT(z, 0.25, 0.75, "t w") Q1 <- qBMT(F1, 0.25, 0.75, "t w") max(abs(z - Q1)) f1 <- dBMT(z, 0.25, 0.75, "t w") r1 <- rBMT(100, 0.25, 0.75, "t w") layout(matrix(c(1,2,1,3), 2, 2)) hist(r1, freq = FALSE, xlim = c(0,1)) lines(z, f1) plot(z, F1, type="l") plot(F1, Q1, type="l") # BMT on [0,1] with asymmetry coefficient equal to 0.5 and # steepness coefficient equal to 0.5 F2 <- pBMT(z, 0.5, 0.5, "a-s") Q2 <- qBMT(F2, 0.5, 0.5, "a-s") f2 <- dBMT(z, 0.5, 0.5, "a-s") r2 <- rBMT(100, 0.5, 0.5, "a-s") max(abs(f1 - f2)) max(abs(F1 - F2)) max(abs(Q1 - Q2)) # BMT on [-1.783489, 3.312195] with # left tail weight equal to 0.25 and # right tail weight equal to 0.75 x <- seq(-1.783489, 3.312195, length.out = 100) F3 <- pBMT(x, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") Q3 <- qBMT(F3, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") max(abs(x - Q3)) f3 <- dBMT(x, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") r3 <- rBMT(100, 0.25, 0.75, "t w", -1.783489, 3.312195, "c-d") layout(matrix(c(1,2,1,3), 2, 2)) hist(r3, freq = FALSE, xlim = c(-1.783489,3.312195)) lines(x, f3) plot(x, F3, type="l") plot(F3, Q3, type="l") # BMT with mean equal to 0, standard deviation equal to 1, # asymmetry coefficient equal to 0.5 and # steepness coefficient equal to 0.5 f4 <- dBMT(x, 0.5, 0.5, "a-s", 0, 1, "l-s") F4 <- pBMT(x, 0.5, 0.5, "a-s", 0, 1, "l-s") Q4 <- qBMT(F4, 0.5, 0.5, "a-s", 0, 1, "l-s") r4 <- rBMT(100, 0.5, 0.5, "a-s", 0, 1, "l-s") max(abs(f3 - f4)) max(abs(F3 - F4)) max(abs(Q3 - Q4))