VGAMextra (version 0.0-6)

maxwellQlink: Link functions for the quantiles of several 1--parameter continuous distributions

Description

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

Usage

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

Value

For deriv = 0, the maxwellQlink transformation of

theta, when inverse = FALSE. If inverse = TRUE, then the inverse given by 2*qgamma(p, 1.5) / theta^2 is returned.

For deriv = 1, this function returns the derivative

\(d\)

eta / \(d\)

theta, if inverse = FALSE. Else, the reciprocal

\(d\)

theta / \(d\)

eta as a function of theta.

If deriv = 2, then the second order derivatives in terms of

theta are accordingly returned.

Arguments

theta

Numeric or character. See below for further details.

p

Numeric. A single value between 0 and 1. It is the \(p\)--quantile to be modeled by this link function.

bvalue, inverse, deriv, short, tag

See Links.

Author

V. Miranda and Thomas W. Yee.

Details

In the VGLM/VGAM quantile regression context, this link function can be used to model any \(p\)--quantile of the Maxwell distribition. It is the maxwellQlink transformation given by $$\sqrt{ 2~\textrm{\texttt{qgamma(p, 1.5) / a.}}}$$ Here, \(a\) is positive parameter as in maxwell whereas qgamma is the quantile function of the gamma distribution.

Numerical values of \(a\) or \(p\) out of range will result in Inf, -Inf, NA or NaN correspondingly.

In particular, arguments inverse and deriv are disregarded if theta is character. Also, if inverse = TRUE and deriv = 0, then argument theta becomes \(\eta\). See Links for further details about this.

See Also

maxwell, Links.

Examples

  ## E1. maxwellQlink() and its inverse ##
   p <- 0.25        ## Modeling the first quartile
   my.a <- seq(0, 5, by = 0.1)[-1]
   max(my.a - maxwellQlink(maxwellQlink(my.a, p = p), p = p, inverse  =TRUE)) ## Zero
    
  ## E2. The first two values are negative, NaN is returned ##
    maxwellQlink(theta = c(-0.15, -0.10, 0.25, 0.35) , p = p, inverse  = FALSE)
    maxwellQlink(theta = c(-0.15, -0.10, 0.25, 0.35) , p = p, inverse  = TRUE)

  ## E3. Plot of the maxwellQlink() and its inverse                    ##
  ## Note, inverse = TRUE implies that argument 'theta' becomes 'eta'. ##
   
  # \donttest{
   #--- THE LINK
   
   plot(maxwellQlink(theta = my.a, p = p) ~ my.a,
        type = "l", col = "blue", lty = "dotted", lwd = 3,
        xlim = c(-0.1, 10), ylim = c(-0.1, 5), las = 1,
        main = c("Blue is maxwellQlink(), green is the inverse"),
        ylab = "eta = maxwellQlink", xlab = "theta")
   abline(h = 0, v = 0, lwd = 2)
    
   #--- THE INVERSE
   lines(my.a, maxwellQlink(theta = my.a, p = p,inv = TRUE), 
          col = "green", lwd = 2, lty = "dashed")
   lines(my.a, my.a)     # Tracing the identity function for double--check
   # }