BMT {BMT} R Documentation

## 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

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.

### Examples

# 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))



[Package BMT version 0.1.0.3 Index]