# Laplace Regression Family Function Maximum likelihood estimation of the 2-parameter classical Laplace distribution. ```r laplace(llocation = "identitylink", lscale = "loglink", ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale") ``` ## Arguments - `llocation, lscale`: Character. Parameter link functions for location parameter $a$ and scale parameter $b$. See `Links` for more choices. - `ilocation, iscale`: Optional initial values. If given, it must be numeric and values are recycled to the appropriate length. The default is to choose the value internally. - `imethod`: Initialization method. Either the value 1 or 2. - `zero`: See `CommonVGAMffArguments` for information. ## Details The Laplace distribution is often known as the **double-exponential** distribution and, for modelling, has heavier tail than the normal distribution. The Laplace density function is $$ f(y) = \frac{1}{2b} \exp \left( - \frac{|y-a|}{b}\right)%f(y) = (1/(2b)) exp( -|y-a|/b ) $$ where $-Inf0$. Its mean is $a$ and its variance is $2b^2$. This parameterization is called the **classical Laplace distribution** by Kotz et al. (2001), and the density is symmetric about $a$. For `y ~ 1` (where `y` is the response) the maximum likelihood estimate (MLE) for the location parameter is the sample median, and the MLE for $b$ is `mean(abs(y-location))` (replace location by its MLE if unknown). ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## References Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). **The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance**, Boston: Birkhauser. ## Author(s) T. W. Yee ## Warning This family function has not been fully tested. The MLE regularity conditions do **not** hold for this distribution, therefore misleading inferences may result, e.g., in the `summary` and `vcov` of the object. Hence this family function might be withdrawn from `VGAM` in the future. ## Note This family function uses Fisher scoring. Convergence may be slow for non-intercept-only models; half-stepping is frequently required. ## See Also `rlaplace`, `alaplace2` (which differs slightly from this parameterization), `exponential`, `median`. ## Examples ```r ldata <- data.frame(y = rlaplace(nn <- 100, 2, scale = exp(1))) fit <- vglm(y ~ 1, laplace, ldata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) with(ldata, median(y)) ldata <- data.frame(x = runif(nn <- 1001)) ldata <- transform(ldata, y = rlaplace(nn, 2, scale = exp(-1 + 1*x))) coef(vglm(y ~ x, laplace(iloc = 0.2, imethod = 2, zero = 1), ldata, trace = TRUE), matrix = TRUE) ```

laplace function