BMTcentral function

The BMT Distribution Descriptive Measures - Central Tendency.

Mean, median and mode 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.

BMTmean(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTmedian(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

BMTmode(p3, p4, type.p.3.4 = "t w", p1 = 0, p2 = 1, type.p.1.2 = "c-d")

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.

Returns

BMTmean gives the mean, BMTmedian the median and BMTmode the mode for the BMT distribution.

The arguments are recycled to the length of the result. Only the first elements of type.p.3.4 and type.p.1.2 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

# BMT on [0,1] with left tail weight equal to 0.25 and 
# right tail weight equal to 0.75
BMTmean(0.25, 0.75, "t w")
BMTmedian(0.25, 0.75, "t w")
BMTmode(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and 
# steepness coefficient equal to 0.75
BMTmean(0.5, 0.5, "a-s")
BMTmedian(0.5, 0.5, "a-s")
BMTmode(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
BMTmean(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTmedian(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTmode(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 0.75
BMTmean(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTmedian(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTmode(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.

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

BMTdispersion, BMTskewness, BMTkurtosis, BMTmoments for other descriptive measures or moments.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co