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

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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")

`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

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.

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.

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

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.

# 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))