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

Description Usage Arguments Details Value Author(s) References See Also Examples

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

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)

`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

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).

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.

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

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.

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)