VGAMextra (version 0.0-7)

weibullMlink: Link functions for the mean of 2--parameter continuous distributions: The Weibull distribution.

Description

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

Usage

weibullMlink(theta, shape = NULL, wrt.param = NULL,
                    bvalue = NULL, inverse = FALSE,
                    deriv = 0, short = TRUE, tag = FALSE)

Value

For deriv = 0, the weibmeanlnik transformation of

theta, i.e., $\beta$, when inverse = FALSE. If inverse = TRUE, then $\theta$ becomes $\eta$, and the inverse,

$\exp (theta-\log \mathrm{\Gamma }(1+1/\alpha )),$

for given

$\alpha$, is returned.

When deriv = 1

theta becomes

$\theta =(\beta ,\alpha )=({\theta }_1,{\theta }_2)$, and

$\eta =({\eta }_1,{\eta }_2)$ with

${\eta }_2=\log \alpha$, and the argument wrt.param must be considered:

A) If inverse = FALSE, then

$d$

eta1 / $d$

$\beta$ is returned when

wrt.param = 1, and

$d$

eta1 / $d$

$\alpha$ if

wrt.param = 2.

B) For inverse = TRUE, this function returns

$d$

$\beta$ / $d$

eta1 and

$d$

$\alpha$ / $d$

eta1 conformably arranged in a matrix, if wrt.param = 1, as a function of ${\theta }_i$, $i=1,2$. When wrt.param = 2, a matrix with columns

$d\beta$ / $d$

eta2 and

$d\alpha$ / $d$

eta2

is returned.

For deriv = 2, the second derivatives in terms of theta are likewise returned.

Arguments

theta

Numeric or character. This is $\theta$ ('scale' parameter) but it may be $\eta$ depending on the other parameters. See below for further details.

shape

The shape parameter. Required for this link to work. See weibullRff.

wrt.param

Positive integer, either $1$ or $2$. The partial derivatives are computed with respect to one of the two linear predictors involved with this link. Further details listed below.

bvalue, inverse, deriv, short, tag

See Links.

Author

V. Miranda and Thomas W. Yee.

Details

This is the link for the mean of the 2--parameter Weibull distribution, also known as the weibullMlink transformation. It can only be used within weibullRff and is defined as $$\mathtt{w}\mathtt{e}\mathtt{i}\mathtt{b}\mathtt{u}\mathtt{l}\mathtt{l}\mathtt{M}\mathtt{l}\mathtt{i}\mathtt{n}\mathtt{k}\mathtt{(}\beta \mathtt{;}\alpha \mathtt{)}\mathtt{=}\eta \mathtt{(}\beta \mathtt{;}\alpha \mathtt{)}\mathtt{=}\log \mathtt{[}\beta \cdot \mathtt{\Gamma }\mathtt{(}\mathtt{1}\mathtt{+}\mathtt{1}\mathtt{/}\alpha \mathtt{)}\mathtt{]}\mathtt{,}$$ for given $\alpha$ ('shape' parameter) where $\beta >0$ is the scale parameter. weibullMlink is expressly a function of $\beta$, i.e. $\theta$, therefore $\alpha$ (shape) must be entered at every call.

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

See Also

weibullQlink, weibullRff, weibullR, lgamma, Links.

Examples

    eta <- seq(-3, 3, by = 0.1) # this is eta = log(mu(b, a)).
    shape  <- exp(1)    # 'shape' argument.
 
 ## E1. Get 'scale' values with A WARNING (not the same length)!
   theta <- weibullMlink(theta = eta, shape = shape, inverse = TRUE)  # Scale
   
 if (FALSE) {
 ## E2. Plot theta vs. eta, 'shape' fixed.
 plot(theta, eta, type = "l", ylab = "", col = "blue",
      main = paste0("weibullMlink(theta; shape = ",
                    round(shape, 3), ")"))
  abline(h = -3:3, v = 0, col = "gray", lty = "dashed")
 }
 
 ## E3. weibullMlink() and its inverse ##
    etabis  <- weibullMlink(theta = theta, shape = shape, inverse = FALSE)
    summary(eta - etabis)     # Should be 0