geometricffMlink {VGAMextra}

Link functions for the mean of 1–parameter discrete distributions: The Geometric Distribution.

Description

Computes the geometricffMlink transformation, including its inverse and the first two derivatives.

Usage

    geometricffMlink(theta, bvalue = NULL, inverse = FALSE, 
                     deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

bvalue, inverse, deriv, short, tag

Details at Links

Details

This is a natural link function to model the mean of the (discret) geometric distribution, geometric, defined as the logarithmm of its mean, i.e.,

\eta = -\log \frac{p}{1 - p} = -{\tt{logit}}(p).

Here, p is the probability of succes, as in geometric.

While this link function can be used to model any parameter lying in (0, 1), it is particularly useful for event-rate geometric data where the mean can be written in terms of some rate of events, say \lambda = \lambda(\mathbf{x}), as

\mu = \lambda(\mathbf{x}) t,

and the time t (as \log t) can be easily incorporated in the analysis as an offset.

Under this link function the domain set for p is (0, 1). Hence, values of \rho too close to the extremes, or out of range will result in Inf, -Inf, NA or NaN. Use argument bvalue to adequately replace them before computing the link function.

If theta is a character, arguments inverse and deriv are disregarded.

Value

For deriv = 0, the geometricffMlink transformation of theta when inverse = FALSE. When inverse = TRUE then theta becomes \eta, and exp(-theta) / (exp(-theta) - 1) is returned.

For deriv = 1, d eta / d theta, if inverse = FALSE, else the reciprocal d theta / d eta as a function of theta.

For deriv = 2 the second order derivatives are correspondingly returned.

Warning

Numerical instability may occur if covariates are used leading to values of p out of range. Try to overcome this by using argument bvalue.

Note

This function may return Inf of -Inf for values of p too close to 0 and 1 respectively.

Author(s)

V. Miranda and T. W. Yee

See Also

geometric, Links, logitlink, logffMlink.

Examples

### Example 1  ###
my.probs <- ppoints(100)
geol.inv <-
    geometricffMlink(theta = geometricffMlink(theta = my.probs), # the inverse
                                inverse = TRUE) - my.probs
summary(geol.inv)     ## zero

###  Example 2. Special values of 'prob'  ###
my.probs <- c(-Inf, -2, -1, 0, 0.25, 0.75, 1.0, 5, Inf, NaN, NA) 
rbind(probs = my.probs, 
      geoffMlink = geometricffMlink(theta = my.probs),
      inv.geoffl = geometricffMlink(theta = my.probs, inverse = TRUE))


###  Example 3 Some probability link functions  ###

my.probs <- ppoints(100)

par(lwd = 2)
plot(my.probs, logitlink(my.probs), xlim = c(-0.1, 1.1), ylim = c(-5, 8),
     type = "l", col = "limegreen", 
     ylab = "transformation", las = 1, main = "Some probability link functions")
lines(my.probs, geometricffMlink(my.probs), col = "gray50")
lines(my.probs, logffMlink(my.probs), col = "blue")
lines(my.probs, probitlink(my.probs), col = "purple")
lines(my.probs, clogloglink(my.probs), col = "chocolate")
lines(my.probs, cauchitlink(my.probs), col = "tan")
abline(v = c(0.5, 1), lty = "dashed")
abline(v = 0, h = 0, lty = "dashed")
legend(0.1, 8, 
      c("geometricffMlink", "logffMlink","logitlink", "probitlink",
      "clogloglink", "cauchitlink"), 
      col = c("gray50", "blue", "limegreen", "purple", "chocolate", "tan"),
      lwd = 1, cex = 0.5)
par(lwd = 1)