BMTmoments {BMT}

The BMT Distribution Moments, Moment-Generating Function and Characteristic Function.

Description

Any raw, central or standarised 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.

Usage

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 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.

order

order of the moment.

type

type of the moment: raw, central or standardised (default).

method

method to obtain the moment: exact formula or Chebyshev-Gauss quadrature (default).

s

variable for the moment-generating and characteristic functions.

Details

See References.

Value

BMTmoment gives any raw, central or standarised 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.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [e-mail]

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. (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.

Examples

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")