# 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] cjtorresj@unal.edu.co

## 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.

Distributions for other standard distributions. `pBMT` for the BMT distribution and `pBMT.Phi` for the BMT-Phi distribution.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```layout(matrix(1:4, 2, 2)) curve(plogis(x, scale = 1 / 1.70174439), -4, 4, col = "red", lty = 2, ylab = "cdf") curve(pBMT.Psi(x), add = TRUE, col = "blue", lty = 3) legend("topleft", legend = c("logis(0, 1 / 1.70174439)","BMT-Psi(0,1)"), bty = "n", col = c("red","blue"), lty = 2:3) curve(plogis(x, scale = 1 / 1.70174439)-pBMT.Psi(x), -4, 4) curve(qlogis(x, scale = 1 / 1.70174439), col = "red", lty = 2, xlab = "p", ylab = "qf") curve(qBMT.Psi(x), add = TRUE, col = "blue", lty = 3) hist(rBMT.Psi(10000), freq = FALSE, breaks = seq(-4, 4, 0.25), border = "blue") curve(dlogis(x, scale = 1 / 1.70174439), add = TRUE, col = "red", lty = 2) curve(dBMT.Psi(x), add = TRUE, col = "blue", lty = 3) ```