expQlink: 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 $\theta$ although may be $\eta$ depending on the other parameters. See below for further details.

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

bvalue, inverse, deriv, short, tag

See Links.

Value

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

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

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

Details

This is a link function to model any fixed quantile, say $\xi_p$, of the exponential distribution. It is called the expQlink transformation and is defined as $$ \log (1 - p)^{-1/\lambda},$$ where $\lambda$ is positive as in exponential.

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

Examples

Run this code

# NOT RUN {
 ## 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)
    
# }
# NOT RUN {
  ##  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"))
  
# }