BMT.Psi {BMT}

The BMT-Psi 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

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 \le 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 \kappa_l = \kappa_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] [e-mail]

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.

See Also

Distributions for other standard distributions. pBMT for the BMT distribution and pBMT.Phi for the BMT-Phi distribution.

Examples

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)