Bridge: The Bridge Distribution

Description

Density, distribution function, quantile function and random generation for the bridge distribution with parameter phi. See Wang and Louis (2003).

Usage

dbridge(x, phi = 1/2, log = FALSE)
pbridge(q, phi = 1/2, lower.tail = TRUE, log.p = FALSE)
qbridge(p, phi = 1/2, lower.tail = TRUE, log.p = FALSE)
rbridge(n, phi = 1/2)

Value

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.

Arguments

x, q: vector of quantiles.
phi: phi parameter. The phi must be between 0 and 1. A phi 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 $P[X \le x]$, otherwise, $P[X > x]$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If phi is omitted, the default value 1/2 is assumed.

The Bridge distribution parameterized by phi has distribution function $$F(q) = 1 - 1/(pi*phi) * (pi/2 - atan( (exp(phi*q) + cos(phi*pi)) / sin(phi*pi) ))$$ and density $$f(x) = 1/(2*pi) * sin(phi*pi) / (cosh(phi*x) + cos(phi*pi)).$$

The mean is $\mu$ and the variance is $\pi^2 (\phi^{-2} - 1) / 3 $.

References

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>

Examples

Run this code

  ## Confirm unit variance for phi = 1/sqrt(1+3/pi^2)
  var(rbridge(1e5, phi = 1/sqrt(1+3/pi^2)))  # approximately 1

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