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

## Link functions for the mean of 1–parameter continuous distributions: The exponential distribution.

### Description

Computes the expMlink transformation, its inverse and the first two derivatives.

### Usage

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


### Arguments

 theta Numeric or character. This is \theta although may be \eta depending on the other parameters. See below for further details. location This is a known location parameter. Same as location in exponential. bvalue, inverse, deriv, short, tag See Links.

### Details

This is a link function to model the mean of the exponential distribution, exponential. It is defined as

\eta = \log (A + \lambda^{-1}),

where \lambda > 0 is a rate parameter and A is a known location parameter, same as exponential.

Numerical values of \lambda out of range may result in Inf, -Inf, NA or NaN.

### Value

For deriv = 0, the expMlink transformation of theta when inverse = FALSE. If inverse = TRUE, then the inverse exp(theta - A)^(-1).

For deriv = 1, d eta / d theta when inverse = FALSE. If inverse = TRUE, then d theta / d eta as a function of theta.

Similarly, when deriv = 2, the second derivatives in terms of theta are returned.

### Note

Numerical instability may occur for values theta too close to zero. Use argument bvalue to replace them before computing the 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.

exponential, Links.

### Examples

 ##  E1. Modelling the mean of the exponential distribution  ##
set.seed(17010402)
nn <- 100
edata <- data.frame(x2 = runif(nn) - 0.5, x3 = runif(nn) - 0.5)
edata <- transform(edata, eta = 0.2 - 0.7 * x2 + 1.9 * x3)

#----- The mean is a function of 'x2' and 'x3' ------#
edata <- transform(edata, rate = expMlink(eta, inverse = TRUE))

edata <- transform(edata, y = rexp(nn, rate = rate))
with(edata, stem(y))
with(edata, hist(y))

data = edata, zero = NULL, trace = TRUE, crit = "log")
coef(exp.fit, matrix = TRUE)
summary(exp.fit)

## E2. expMlink() and its inverse ##
theta <- 0.1 + 1:5
location <- 1.5