# BMT.Psi: The BMT-Psi Distribution. In BMT: The BMT Distribution

## Description

Density, distribution function, quantile function, random number generation for the BMT-Psi distribution with mean equal to mean and standard deviation equal to sd.

## Usage

 1 2 3 4 5 6 7 dBMT.Psi(x, mean = 0, sd = 1, log = FALSE) pBMT.Psi(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE) qBMT.Psi(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE) rBMT.Psi(n, mean = 0, sd = 1)

## Arguments

 x, q vector of quantiles. mean vector of means. sd vector of standard deviations. 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

If mean or sd are not specified they assume the default values of 0 and 1, respectively.

The BMT-Psi distribution is the BMT distribution with κ_l = κ_r = 0.63355781127887611515. The BMT-Psi cumulative distribution function (cdf) is the closest BMT cdf to the logistic cdf with scale = 1 / d and d = 1.70174439 (Camilli, 1994, p. 295).

## Value

dBMT.Psi gives the density, pBMT.Psi the distribution function, qBMT.Psi the quantile function, and rBMT.Psi generates random deviates.

The length of the result is determined by n for rBMT.Psi, 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.

sd <= 0 is an error and returns NaN.

## Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [email protected]

## References

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.

Camilli, G. (1994). Teacher's corner: origin of the scaling constant d= 1.7 in item response theory. Journal of Educational Statistics, 19(3), 293-295.