inv.chisqMeanlink: Link functions for the mean of 1--parameter continuous distributions: The inverse chi--squared distribution.

Description

Computes the inv.chisqMeanlink transformation, its inverse and the first two derivatives.

Usage

inv.chisqMeanlink(theta, bvalue = NULL, inverse = FALSE,
                          deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. This is $\theta$ by default but may be $\eta$ depending on the other parameters. See below for further details.

bvalue, inverse, deriv, short, tag

See Links.

Value

For deriv = 0, the inv.chisqMeanlink transformation of theta when inverse = FALSE. If inverse = TRUE, then the inverse exp(-theta) + 2.

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

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

Details

This function arises as the link to model the mean of the inverse chi--squared distribution, inv.chisq.

It is defined as $$ \eta = -\log ( df - 2),$$ where $df$ denotes the (non--negative) degrees of freedom, as in inv.chisq.

Notice, however, that $df > 2$ is required for the mean of this distribution to be real. Consequently, the domain set for df for this link function is $(2, \infty)$.

Numerical values of $df$ out of range will result in NA or NaN.

Examples

Run this code

# NOT RUN {
 ##  E1. Modelling the mean of the exponential distribution  ##
    set.seed(17010502)
    dof <- 2.5 
    isq.data <- data.frame(x2 = runif(100, 0, 1))
    isq.data <- transform(isq.data, y = rinv.chisq(n = 100, df = dof + x2))
    
    
# }
# NOT RUN {
    hist(isq.data$y)
    
# }
# NOT RUN {
    
    fit.inv <- vglm(y ~ x2, family = inv.chisq(link = "inv.chisqMeanlink"), 
                    data = isq.data, trace = TRUE )
    coef(fit.inv, matrix = TRUE)
    summary(fit.inv)

  ## E3. Special values in a matrix ##
    (theta <- matrix(c(Inf, -Inf, NA, NaN, 1 , 2, 3, 4), ncol = 4, nrow = 2))
    inv.chisqMeanlink(theta = theta)   ## NaNs for df = theta <= 2 
 
  ## E2. inv.chisqMeanlink() and its inverse ##
    theta <- 0.1 + 1:5  # dof = df
    my.diff <- theta - inv.chisqMeanlink(inv.chisqMeanlink(theta = theta), inverse  =TRUE)
    summary(my.diff)     # Zero

# }