BMTkurtosis {BMT} R Documentation

## The BMT Distribution Descriptive Measures - Kurtosis.

### Description

Kurtosis and steepness coefficient 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

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

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

### Arguments

 `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.

See References.

### Value

`BMTkurt` gives the Pearson's kurtosis and `BMTsteep` the proposed steepness coefficient 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`.

### Author(s)

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

### 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.

`BMTcentral`, `BMTdispersion`, `BMTskewness`, `BMTmoments` for other descriptive measures or moments.

### Examples

```# BMT on [0,1] with left tail weight equal to 0.25 and
# right tail weight equal to 0.75
BMTkurt(0.25, 0.75, "t w")
BMTsteep(0.25, 0.75, "t w")

# BMT on [0,1] with asymmetry coefficient equal to 0.5 and
# steepness coefficient equal to 0.75
BMTkurt(0.5, 0.5, "a-s")
BMTsteep(0.5, 0.5, "a-s")

# domain or location-scale parameters do not affect
# the skewness and the asymmetry coefficient

# BMT on [-1.783489,3.312195] with
# left tail weight equal to 0.25 and
# right tail weight equal to 0.75
BMTkurt(0.25, 0.75, "t w", -1.783489, 3.312195, "c-d")
BMTsteep(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
BMTkurt(0.5, 0.5, "a-s", 0, 1, "l-s")
BMTsteep(0.5, 0.5, "a-s", 0, 1, "l-s")
```

[Package BMT version 0.1.0.3 Index]