# Exponential Distribution Maximum likelihood estimation for the exponential distribution. ```r exponential(link = "loglink", location = 0, expected = TRUE, type.fitted = c("mean", "percentiles", "Qlink"), percentiles = 50, ishrinkage = 0.95, parallel = FALSE, zero = NULL) ``` ## Arguments - `link`: Parameter link function applied to the positive parameter $rate$. See `Links` for more choices. - `location`: Numeric of length 1, the known location parameter, $A$, say. - `expected`: Logical. If `TRUE` Fisher scoring is used, otherwise Newton-Raphson. The latter is usually faster. - `ishrinkage, parallel, zero`: See `CommonVGAMffArguments` for information. - `type.fitted, percentiles`: See `CommonVGAMffArguments` for information. ## Details The family function assumes the response $Y$ has density $$ f(y) = \lambda \exp(-\lambda (y-A))%f(y) = rate * exp(-rate * (y-A)) $$ for $y > A$, where $A$ is the known location parameter. By default, $A=0$. Then $E(Y) = A + 1/rate$ and $Var(Y) = 1/rate^2$. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). **Statistical Distributions**, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition. ## Author(s) T. W. Yee ## Note Suppose $A = 0$. For a fixed time interval, the number of events is Poisson with mean $rate$ if the time between events has a geometric distribution with mean $1/rate$. The argument `rate` in `exponential` is the same as `rexp` etc. The argument `lambda` in `rpois` is somewhat the same as `rate` here. ## See Also `amlexponential`, `gpd`, `laplace`, `expgeometric`, `explogff`, `poissonff`, `mix2exp`, `freund61`, `simulate.vlm`, `Exponential`. ## Examples ```r edata <- data.frame(x2 = runif(nn <- 100) - 0.5) edata <- transform(edata, x3 = runif(nn) - 0.5) edata <- transform(edata, eta = 0.2 - 0.7 * x2 + 1.9 * x3) edata <- transform(edata, rate = exp(eta)) edata <- transform(edata, y = rexp(nn, rate = rate)) with(edata, stem(y)) fit.slow <- vglm(y ~ x2 + x3, exponential, data = edata, trace = TRUE) fit.fast <- vglm(y ~ x2 + x3, exponential(exp = FALSE), data = edata, trace = TRUE, crit = "coef") coef(fit.slow, mat = TRUE) summary(fit.slow) # Compare results with a GPD. Has a threshold. threshold <- 0.5 gdata <- data.frame(y1 = threshold + rexp(n = 3000, rate = exp(1.5))) fit.exp <- vglm(y1 ~ 1, exponential(location = threshold), data = gdata) coef(fit.exp, matrix = TRUE) Coef(fit.exp) logLik(fit.exp) fit.gpd <- vglm(y1 ~ 1, gpd(threshold = threshold), data = gdata) coef(fit.gpd, matrix = TRUE) Coef(fit.gpd) logLik(fit.gpd) ```

exponential function