expQlink {VGAMextra}R Documentation

Link functions for the quantiles of several 1–parameter continuous distributions.

Description

Computes the expQlink transformation, its inverse and the first two derivatives.

Usage

  expQlink(theta, p = stop("Argument 'p' must be entered."),
           bvalue = NULL, inverse = FALSE,
           deriv = 0, short = TRUE, tag = FALSE) 

Arguments

theta

Numeric or character. This is θ\theta although may be η\eta depending on the other parameters. See below for further details.

p

Numeric. A prespecified number between 0 and 1. The particular pp–quantile to be modelled. For example, p=0.5p = 0.5 means that the median is considered by this link function.

bvalue, inverse, deriv, short, tag

See Links.

Details

This is a link function to model any fixed quantile, say ξp\xi_p, of the exponential distribution. It is called the expQlink transformation and is defined as

log(1p)1/λ, \log (1 - p)^{-1/\lambda},

where λ\lambda is positive as in exponential.

Numerical values of λ\lambda or pp out of range may result in Inf, -Inf, NA or NaN.

Value

With deriv = 0, the expQlink transformation of theta for prespecified pp when inverse = FALSE. If inverse = TRUE, then the inverse -log(1 - p)/theta.

For deriv = 1, this link function returns dd eta / dd theta when inverse = FALSE. If inverse = TRUE, then dd theta / dd eta as a function of theta.

Similarly, when deriv = 2, the second derivatives in terms of theta are returned.

Note

Numerical instability may occur for values theta too close to zero. Use argument bvalue to replace them before computing the link.

If theta is character, then arguments inverse and deriv are ignored. See Links for further details.

Author(s)

V. Miranda and Thomas W. Yee.

See Also

exponential, Links.

Examples

 ## E1. expQlink() and its inverse ##
    p <- 0.25            # Modelling the first quartile.
    my.theta <- seq(0, 5, by = 0.1)[-1]
    my.diff <- my.theta - expQlink(expQlink(my.theta, p = p), p = p, inverse  =TRUE)
    summary(my.diff)     # Zero


  ## E2. Special values ##
    expQlink(theta = c(Inf, -Inf, NA, NaN), p = p)
    

  ##  E3. Plot of expQlink() for different quantiles  ##
    plot(expQlink(my.theta, p = p) ~ my.theta, 
         type = "l", lty = "dotted", col = "blue", lwd = 2,
         main = "expQink(p) transformation", xlab = "theta", ylab = "expQLink",
         xlim = c(-0.5, 5), ylim = c(-0.5, 5))
    abline(h = 0, v = 0, lwd = 2)
    lines(my.theta, expQlink(my.theta, p = 0.50),  col = "green", lty = "dotted", lwd = 2)
    lines(my.theta, expQlink(my.theta, p = 0.75),  col = "red", lty = "dotted", lwd = 2)
    legend(2, 4,  c("p = 0.25", "p = 0.50", "p = 0.75"), col = c("blue", "green", "red"),
           lwd = c(2, 2, 2), lty = c("dotted", "dotted", "dotted"))
  

[Package VGAMextra version 0.0-6 Index]