Description Usage Arguments Details Value Note Source References See Also Examples

Density, distribution function, quantile function and random generation for
the bridge distribution with parameter `scale`

. See Wang and Louis (2003).

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`x, q` |
vector of quantiles. |

`scale` |
scale parameter. The scale must be between 0 and 1. A scale of 1/sqrt(1+3/pi^2) gives unit variance. |

`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 `scale`

is omitted, the default
value `1/2`

is assumed.

The Bridge distribution parameterized by
`scale`

has distribution function

*F(q) = 1 - 1/(pi*scale) * (pi/2 - atan( (exp(scale*q) + cos(scale*pi)) / sin(scale*pi) ))*

and density

*f(x) = 1/(2*pi) * sin(scale*pi) / (cosh(scale*x) + cos(scale*pi)).*

The mean is *0* and the variance is
*pi^2 * (scale^{-2} - 1) / 3 *.

`dbridge`

gives the density, `pbridge`

gives the
distribution function, `qbridge`

gives the quantile function, and
`rbridge`

generates random deviates.

The length of the result is determined by `n`

for `rbridge`

, 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.

Consult the vignette for some figures comparing the normal, logistic, and bridge distributions.

`[dpq]bridge`

are calculated directly from the definitions.

`rbridge`

uses inversion.

Wang, Z. and Louis, T.A. (2003) Matching conditional and marginal shapes in binary random intercept models using
a bridge distribution function. *Biometrika*, 90(4), 765-775. <DOI:10.1093/biomet/90.4.765>

See also:

Swihart, B.J., Caffo, B.S., and Crainiceanu, C.M. (2013). A Unifying Framework for Marginalized Random-Intercept Models of Correlated Binary Outcomes. *International Statistical Review*, 82 (2), 275-295 1-22. <DOI: 10.1111/insr.12035>

Griswold, M.E., Swihart, B.J., Caffo, B.S and Zeger, S.L. (2013). Practical marginalized multilevel models. Stat, 2(1), 129-142. <DOI: 10.1002/sta4.22>

Heagerty, P.J. (1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics, 55(3), 688-698. <DOI: 10.1111/j.0006-341X.1999.00688.x>

Heagerty, P.J. and Zeger, S.L. (2000). Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors). Stat. Sci., 15(1), 1-26. <DOI: 10.1214/ss/1009212671>

Distributions for other standard distributions.

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