VGAM (version 1.0-1)

gev: Generalized Extreme Value Distribution Family Function

Description

Maximum likelihood estimation of the 3-parameter generalized extreme value (GEV) distribution.

Usage

gev(llocation = "identitylink", lscale = "loge", lshape = logoff(offset = 0.5),
    percentiles = c(95, 99), iscale=NULL, ishape = NULL,
    imethod = 1, gshape = c(-0.45, 0.45), tolshape0 = 0.001,
    type.fitted = c("percentiles", "mean"), giveWarning = TRUE,
    zero = c("scale", "shape"))
egev(llocation = "identitylink", lscale = "loge", lshape = logoff(offset = 0.5),
     percentiles = c(95, 99), iscale=NULL,  ishape = NULL,
     imethod = 1, gshape = c(-0.45, 0.45), tolshape0 = 0.001,
     type.fitted = c("percentiles", "mean"), giveWarning = TRUE,
     zero = c("scale", "shape"))

Arguments

llocation, lscale, lshape

Parameter link functions for $\mu$, $\sigma$ and $\xi$ respectively. See Links for more choices.

For the shape parameter, the default logoff link has

percentiles

Numeric vector of percentiles used for the fitted values. Values should be between 0 and 100. This argument is ignored if type.fitted = "mean".

type.fitted

See CommonVGAMffArguments for information. The default is to use the percentiles argument. If "mean" is chosen, then the mean $\mu + \sigma (\Gamma(1-\xi)-1) /

iscale, ishape

Numeric. Initial value for $\sigma$ and $\xi$. A NULL means a value is computed internally. The argument ishape is more important than the other two because they are initialized from the initial $\xi$. If a failure to con

imethod

Initialization method. Either the value 1 or 2. Method 1 involves choosing the best $\xi$ on a course grid with endpoints gshape. Method 2 is similar to the method of moments. If both methods fail try using ishape.

gshape

Numeric, of length 2. Range of $\xi$ used for a grid search for a good initial value for $\xi$. Used only if imethod equals 1.

tolshape0, giveWarning

Passed into dgev when computing the log-likelihood.

zero

A specifying which linear/additive predictors are modelled as intercepts only. The values can be from the set {1,2,3} corresponding respectively to $\mu$, $\sigma$, $\xi$. If zero = NULL then all linear/additive predictors are modelle

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Currently, if an estimate of $\xi$ is too close to 0 then an error will occur for gev() with multivariate responses. In general, egev() is more reliable than gev().

Fitting the GEV by maximum likelihood estimation can be numerically fraught. If $1 + \xi (y-\mu)/ \sigma \leq 0$ then some crude evasive action is taken but the estimation process can still fail. This is particularly the case if vgam with s is used; then smoothing is best done with vglm with regression splines (bs or ns) because vglm implements half-stepsizing whereas vgam doesn't (half-stepsizing helps handle the problem of straying outside the parameter space.)

Details

The GEV distribution function can be written $$G(y) = \exp( -[ (y-\mu)/ \sigma ]_{+}^{- 1/ \xi})$$ where $\sigma > 0$, $-\infty < \mu < \infty$, and $1 + \xi(y-\mu)/\sigma > 0$. Here, $x_+ = \max(x,0)$. The $\mu$, $\sigma$, $\xi$ are known as the location, scale and shape parameters respectively. The cases $\xi>0$, $\xi<0$, $\xi="0$" correspond="" to="" the="" frechet,="" weibull,="" and="" gumbel="" types="" respectively.="" it="" can="" be="" noted="" that="" (or="" type="" i)="" distribution="" accommodates="" many="" commonly-used="" distributions="" such="" as="" normal,="" lognormal,="" logistic,="" gamma,="" exponential="" weibull.<="" p="">

For the GEV distribution, the $k$th moment about the mean exists if $\xi < 1/k$. Provided they exist, the mean and variance are given by $\mu+\sigma{ \Gamma(1-\xi)-1}/ \xi$ and $\sigma^2 { \Gamma(1-2\xi) - \Gamma^2(1-\xi) } / \xi^2$ respectively, where $\Gamma$ is the gamma function.

Smith (1985) established that when $\xi > -0.5$, the maximum likelihood estimators are completely regular. To have some control over the estimated $\xi$ try using lshape = logoff(offset = 0.5), say, or lshape = extlogit(min = -0.5, max = 0.5), say.

References

Yee, T. W. and Stephenson, A. G. (2007) Vector generalized linear and additive extreme value models. Extremes, 10, 1--19.

Tawn, J. A. (1988) An extreme-value theory model for dependent observations. Journal of Hydrology, 101, 227--250.

Prescott, P. and Walden, A. T. (1980) Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika, 67, 723--724.

Smith, R. L. (1985) Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72, 67--90.

See Also

rgev, gumbel, egumbel, guplot, rlplot.egev, gpd, weibullR, frechet, extlogit, oxtemp, venice.

Examples

# Multivariate example
fit1 <- vgam(cbind(r1, r2) ~ s(year, df = 3), gev(zero = 2:3),
             data = venice, trace = TRUE)
coef(fit1, matrix = TRUE)
head(fitted(fit1))
par(mfrow = c(1, 2), las = 1)
plot(fit1, se = TRUE, lcol = "blue", scol = "forestgreen",
     main = "Fitted mu(year) function (centered)", cex.main = 0.8)
with(venice, matplot(year, depvar(fit1)[, 1:2], ylab = "Sea level (cm)",
     col = 1:2, main = "Highest 2 annual sea levels", cex.main = 0.8))
with(venice, lines(year, fitted(fit1)[,1], lty = "dashed", col = "blue"))
legend("topleft", lty = "dashed", col = "blue", "Fitted 95 percentile")

# Univariate example
(fit <- vglm(maxtemp ~ 1, egev, data = oxtemp, trace = TRUE))
head(fitted(fit))
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
vcov(fit, untransform = TRUE)
sqrt(diag(vcov(fit)))  # Approximate standard errors
rlplot(fit)