# Gumbel location-scale model family The `gumbls` family implements Gumbel location scale additive models in which the location and scale parameters (see details) can depend on additive smooth predictors. Useable only with `gam`, the linear predictors are specified via a list of formulae. ```r gumbls(link=list("identity","log"),b=-7) ``` ## Arguments - `link`: two item list specifying the link for the location $m$ and log scale parameter $B$. See details for meaning, which may not be intuitive. - `b`: The minumum log scale parameter. ## Returns An object inheriting from class `general.family`. ## Details Let $z = (y - m)exp(-B)$, then the log Gumbel density is $l = -B - z - exp(-z)$. The expected value of a Gumbel r.v. is $m + g exp(B)$ where $g$ is Eulers constant (about 0.57721566). The corresponding variance is $pi^2 exp(2B)/6$. `gumbls` is used with `gam` to fit Gumbel location - scale models parameterized in terms of location parameter $m$ and the log scale parameter $B$. Note that `identity` link for the scale parameter means that the corresponding linear predictor gives $B$ directly. By default the `log` link for the scale parameter simply forces the log scale parameter to have a lower limit given by argument `b`: if $l$ is the linear predictor for the log scale parameter, $B$, then $B = b + log(1+e^l)$. `gam` is called with a list containing 2 formulae, the first specifies the response on the left hand side and the structure of the linear predictor for location parameter, $m$, on the right hand side. The second is one sided, specifying the linear predictor for the lg scale, $B$, on the right hand side. The fitted values for this family will be a two column matrix. The first column is the mean, and the second column is the log scale parameter, $B$. Predictions using `predict.gam` will also produce 2 column matrices for `type` `"link"` and `"response"`. The first column is on the original data scale when `type="response"` and on the log mean scale of the linear predictor when `type="link"`. The second column when `type="response"` is again the log scale parameter, but is on the linear predictor when `type="link"`. ## References Wood, S.N., N. Pya and B. Saefken (2016), Smoothing parameter and model selection for general smooth models. Journal of the American Statistical Association 111, 1548-1575 tools:::Rd_expr_doi("10.1080/01621459.2016.1180986") ## Examples ```r library(mgcv) ## simulate some data f0 <- function(x) 2 * sin(pi * x) f1 <- function(x) exp(2 * x) f2 <- function(x) 0.2 * x^11 * (10 * (1 - x))^6 + 10 * (10 * x)^3 * (1 - x)^10 n <- 400;set.seed(9) x0 <- runif(n);x1 <- runif(n); x2 <- runif(n);x3 <- runif(n); mu <- f0(x0)+f1(x1) beta <- exp(f2(x2)/5) y <- mu - beta*log(-log(runif(n))) ## Gumbel quantile function b <- gam(list(y~s(x0)+s(x1),~s(x2)+s(x3)),family=gumbls) plot(b,pages=1,scale=0) summary(b) gam.check(b) ```

gumbls function