# rayleighMeanlink: Link functions for the mean of 1-parameter continuous... In VGAMextra: Additions and Extensions of the 'VGAM' Package

## Description

The `rayleighMlink` and the `maxwellMlink` transformations, their inverse and the first two derivatives.

## Usage

 ```1 2 3 4 5``` ``` rayleighMlink(theta, bvalue = NULL, inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE) maxwellMlink(theta, bvalue = NULL, inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE) ```

## Arguments

 `theta` Numeric or character. It is theta by default, but it may be eta depending upon other parameters. See `Links` for further details. `bvalue, inverse, deriv, short, tag` See `Links`.

## Details

`rayleighMlink` and `maxwellMlink` are link functions to model the mean of the Rayleigh distirbution, (`rayleigh`), and the mean of the Maxwell distribution, (`maxwell`), respectively.

Both links are somehow defined as the log(theta) plus an offset. Specifcally,

rayleighMlink(b) = log ( b * γ(0.5) / sqrt(2) ),

where b > 0 is a scale parameter as in `rayleigh`; and

maxwellMlink(b) = log ( a^(-1/2) * sqrt(pi/8) ).

Here, a is positive as in `maxwell`.

Non–positive values of a and/or b will result in `NaN`, whereas values too close to zero will return `Inf` or `-Inf`.

## Value

For `deriv = 0`, the corresponding transformation of `theta` when `inverse = FALSE`. If `inverse = TRUE`, then `theta` becomes η, and the inverse transformations

I) `exp(theta) * sqrt`(2) / `gamma(0.5)` for `rayleighMlink`, and

II) 8` * exp(-2 * theta) ` / `gamma(0.5)`^2 for `maxwellMlink`,

are returned.

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.

## Note

Values of a or b out of range, e.g. when covariates involved, may cause numerical instability. Use argument `bvalue` to replace them before computing any link.

If `theta` is character, then arguments `inverse` and `deriv` are ignored. See `Links` for further details.

## Author(s)

V. Miranda and Thomas W. Yee.

`maxwell`, `rayleigh` `Links`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ``` ## The link and its inverse ## theta <- 0.1 + 1:10 eta <- maxwellMlink(maxwellMlink(theta = theta), inverse =TRUE) summary(eta - theta) # Zero eta <- rayleighMlink(rayleighMlink(theta = theta), inverse =TRUE) summary(eta - theta) # Zero ## Modelling the mean of the Maxwell distribution ## set.seed(17010401) rate <- maxwellMlink(theta = 2, inverse = TRUE) # ~ 0.046 mdata <- data.frame(y = rmaxwell(1000, rate = rate )) fit <- vglm(y ~ 1, maxwell(link = "maxwellMlink"), data = mdata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) ```