# pregibon: Pregibon Distribution In glmx: Generalized Linear Models Extended

## Description

Density, distribution function, quantile function and random generation for the Pregibon distribution with parameters `a` and `b`. It is a special case of the generalized Tukey lambda distribution.

## Usage

 ```1 2 3 4``` ```dpregibon(x, a = 0, b = 0, log = FALSE, tol = 1e-12) ppregibon(q, a = 0, b = 0, lower.tail = TRUE, log.p = FALSE, tol = 1e-12) qpregibon(p, a = 0, b = 0, lower.tail = TRUE, log.p = FALSE) rpregibon(n, a = 0, b = 0) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `a, b` distribution parameters. `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]. `tol` numeric tolerance for computation of the distribution function.

## Details

The distribution is a special case of the generalized Tukey lambda distribution and is used by Pregibon (1980) for goodness-of-link testing. See Koenker (2006) and Koenker and Yoon (2009) for more details.

The implementation is based on the corresponding functions for the GeneralisedLambdaDistribution in the gld package (King 2013).

The corresponding link generator is available in the function `pregibon`.

## Value

`dpregibon` gives the probability density function, `ppregibon` gives the cumulative distribution function, `qpregibon` gives the quantile function, and `rpregibon` generates random deviates.

## References

King R (2013). “Estimation and Use of the Generalised (Tukey) Lambda Distribution.” R package version 2.0.1. http://CRAN.R-project.org/package=gld

Koenker R (2006). “Parametric Links for Binary Response.” R News, 6(4), 32–34.

Koenker R, Yoon J (2009). “Parametric Links for Binary Choice Models: A Fisherian-Bayesian Colloquy.” Journal of Econometrics, 152, 120–130.

Pregibon D (1980). “Goodness of Link Tests for Generalized Linear Models.” Journal of the Royal Statistical Society C, 29, 15–23.

GeneralisedLambdaDistribution, `pregibon`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## Koenker & Yoon (2009), Figure 2 par(mfrow = c(3, 3)) pregiboncurve <- function(a, b, from, to, n = 301) { dp <- function(x) dpregibon(x, a = a, b = b) curve(dp, from = from, to = to, n = n, xlab = "", ylab = "", main = paste("a = ", a, ", b = ", b, sep = "")) } pregiboncurve(-0.25, -0.25, -5, 65) pregiboncurve(-0.25, 0, -18, 18) pregiboncurve(-0.25, 0.25, -65, 5) pregiboncurve( 0, -0.25, -4, 22) pregiboncurve( 0, 0, -8, 8) pregiboncurve( 0, 0.25, -22, 4) pregiboncurve( 0.25, -0.25, -2.4,9) pregiboncurve( 0.25, 0, -4, 4) pregiboncurve( 0.25, 0.25, -9, 2.4) par(mfrow = c(1, 1)) ```