# Link functions for the mean of 1--parameter continuous distributions: The exponential distribution. Computes the `expMlink` transformation, its inverse and the first two derivatives. ```r 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 $$ \eta = \log (A + \lambda^{-1}),\eta = log (A + rate^(-1)), $$ where $rate > 0$ is a rate parameter and $A$ is a known location parameter, same as `exponential`. Numerical values of $rate$ out of range may result in `Inf`, `-Inf`, `NA` or `NaN`. ## Returns 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. ## See Also `exponential`, `Links`. ## Examples ```r ## 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)) exp.fit <- vglm(y ~ x2 + x3, exponential(link = "expMlink"), 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 my.diff <- theta - expMlink(expMlink(theta = theta, location = location), location = location, inverse =TRUE) summary(my.diff) # Zero ## E3. Special values in a matrix ## theta <- matrix(c(Inf, -Inf, NA, NaN, 1 , 2), ncol = 3, nrow = 2) expMlink(theta = theta, location = location) ```

expMeanlink function