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

## Link functions for the mean of 1â€“parameter continuous distributions: The Rayleigh and the Maxwell distributions.

### Description

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

### Usage

   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 {\tt{theta}}  plus an offset. Specifcally,

 {\tt{rayleighMlink}}(b) = \log ( b * \gamma(0.5) / sqrt{2} ),

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

 {\tt{maxwellhMlink}}(b) = \log ( a^{-1/2} * sqrt{8 / \pi} ).

Here, a is positive as in maxwell.

Nonâ€“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 \eta, 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.

### Examples

 ##  The link and its inverse ##
theta <- 0.1 + 1:10
summary(eta - theta)     # Zero

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 ))