VGAM (version 1.0-4)

## Description

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

## Usage

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

## Arguments

theta

Numeric or character. See below for further details.

bvalue

Values of `theta` which are less than or equal to 1 can be replaced by `bvalue` before computing the link function value. The component name `bvalue` stands for ``boundary value''. See `Links` for more information.

inverse, deriv, short, tag

Details at `Links`.

## Value

For `deriv = 0`, the log of `theta`, i.e., `log(log(theta))` when `inverse = FALSE`, and if `inverse = TRUE` then `exp(exp(theta))`.

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

Here, all logarithms are natural logarithms, i.e., to base e.

## Details

The log-log link function is commonly used for parameters that are greater than unity. Numerical values of `theta` close to 1 or out of range result in `Inf`, `-Inf`, `NA` or `NaN`.

## References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

`Links`, `loge`, `logoff`.

## Examples

Run this code
```# NOT RUN {
x <- seq(0.8, 1.5, by = 0.1)
loglog(x)  # Has NAs
loglog(x, bvalue = 1.0 + .Machine\$double.eps)  # Has no NAs

x <- seq(1.01, 10, len = 100)
loglog(x)
max(abs(loglog(loglog(x), inverse = TRUE) - x))  # Should be 0
# }
```

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