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`

.

1 2 3 4 5 6 7 |

`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` |
vector of probabilities. |

`n` |
number of observations. If |

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] [email protected]

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)
``` |

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