gamma1Qlink | R Documentation |

Computes the `gamma1Qlink`

transformation, its inverse and the
first two derivatives.

```
gamma1Qlink(theta, p = stop("Argument 'p' must be specified."),
bvalue = NULL, inverse = FALSE,
deriv = 0, short = TRUE, tag = FALSE)
```

`theta` |
Numeric or character. It is |

`p` |
A numeric vector of |

`bvalue, inverse, deriv, short, tag` |
See |

This link function has been specifically designed to model any
`p`

–quantile of the 1–parameter gamma distribution,
`gamma1`

, in the VGLM/VGAM context.
It is defined as

`\eta = \log {\tt{qgamma}}({\tt{p}}, {\tt{shape =}} s),`

where `s`

is a positive shape parameter as in
`gamma1`

, whilst
`{\tt{qgamma()}}`

is the quantile function
`qgamma`

.

The inverse of the `gamma1Qlink`

cannot be expressed in closed
form. Instead, the inverse image, `s_{\eta}`

,
of `\eta`

is numerically approximated by
`newtonRaphson.basic`

.

Numerical values of `s`

or `p`

out of range will
result in `Inf`

, `-Inf`

, `NA`

or `NaN`

correspondingly.

Arguments `inverse`

and `deriv`

are dismissed
if `theta`

is character.

For `deriv = 0`

, the `gamma1Qlink`

transformation of
`theta`

, when `inverse = FALSE`

.
If `inverse = TRUE`

, then
`theta`

becomes `\eta`

, and therefore,
the approximate inverse image of `\eta`

is returned.

For `deriv = 1`

, the partial derivative
`d`

`eta`

/ `d`

`theta`

is returned,
if `inverse = FALSE`

. If `inverse = TRUE`

,
then the reciprocal
`d`

`theta`

/ `d`

`eta`

as a function of `theta`

.

If `deriv = 2`

, then the second order derivatives as a function
of `theta`

.

Numerical instability may occur for values `theta`

too large, or
too close to 0.0. Use argument `bvalue`

to replace them before
computing the link.

V. Miranda and Thomas W. Yee.

`gamma1`

,
`qgamma`

,
`Links`

.

```
## E1. gamma1QLink() and values causing NaNs or out of range ##
p <- 0.75 # The third quartile is of interest.
my.s <- seq(0, 5, by = 0.1)[-1]
max(my.s - gamma1Qlink(gamma1Qlink(my.s, p = p), p = p, inverse =TRUE)) ## Zero
## E2. Special values of theta ##
gamma1Qlink(theta = c(-0.15, -0.10, 0, 1:10) , p = p, inverse = FALSE) ## NaNs
gamma1Qlink(theta = c(-5, -3, 0, 1:10) , p = p, inverse = TRUE) ## Out of range
## E3. Plot of gamma1QLink() and its inverse. ##
# gamma1Qlink()
plot(gamma1Qlink(theta = my.s, p = p) ~ my.s,
type = "l", col = "blue", lty = "dotted", lwd = 3,
xlim = c(-0.1, 5), ylim = c(-5, 15), las = 1,
main = c("Blue is gamma1Qlink(), green is the inverse"),
ylab = "gamma1Qlink transformation", xlab = "theta")
abline(h = 0, v = 0, lwd = 2)
# The inverse
lines(my.s, gamma1Qlink(theta = my.s, p = p, inverse = TRUE),
col = "green", lwd = 2, lty = "dashed")
# The identity function, for double-checking.
lines(my.s, my.s, lty = "dotted")
```

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