VGAM (version 0.7-8)

gev: Generalized Extreme Value Distribution Family Function

Description

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

Usage

gev(llocation = "identity", lscale = "loge", lshape = "logoff",
    elocation = list(), escale = list(),
    eshape = if(lshape=="logoff") list(offset=0.5) else
    if(lshape=="elogit") list(min=-0.5, max=0.5) else list(),
    percentiles = c(95, 99),
    iscale=NULL, ishape = NULL,
    method.init = 1, gshape=c(-0.45, 0.45), tshape0=0.001, zero = 3)
egev(llocation = "identity", lscale = "loge", lshape = "logoff",
     elocation = list(), escale = list(),
     eshape = if(lshape=="logoff") list(offset=0.5) else
     if(lshape=="elogit") list(min=-0.5, max=0.5) else list(),
     percentiles = c(95, 99),
     iscale=NULL,  ishape = NULL,
     method.init=1, gshape=c(-0.45, 0.45), tshape0=0.001, zero = 3)

Arguments

llocation, lscale, lshape

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

elocation, escale, eshape

List. Extra argument for the respective links. See earg in Links for general information. For the shape parameter, if the logoff link is chos

percentiles

Numeric vector of percentiles used for the fitted values. Values should be between 0 and 100. However, if percentiles=NULL, then the mean $\mu + \sigma (\Gamma(1-\xi)-1) / \xi$ is returned, and this is only defined if $\xi

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

method.init

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 method.init equals 1.

tshape0

Positive numeric. Threshold/tolerance value for resting whether $\xi$ is zero. If the absolute value of the estimate of $\xi$ is less than this value then it will be assumed zero and Gumbel derivatives etc. will be used.

zero

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

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 zero 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" and the eshape=list(offset=0.5), say, or lshape="elogit" and eshape=list(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, elogit, oxtemp, venice.

Examples

# Multivariate example
data(venice)
y = as.matrix(venice[,paste("r", 1:10, sep="")])
fit1 = vgam(y[,1:2] ~ s(year, df=3), gev(zero=2:3), venice, trace=TRUE)
coef(fit1, matrix=TRUE)
fitted(fit1)[1:4,]
par(mfrow=c(1,2), las=1)
plot(fit1, se=TRUE, lcol="blue", scol="forestgreen",
     main="Fitted mu(year) function (centered)")
with(venice, matplot(year, y[,1:2], ylab="Sea level (cm)", col=1:2,
     main="Highest 2 annual sealevels + fitted 95 percentile"))
with(venice, lines(year, fitted(fit1)[,1], lty="dashed", col="blue"))


# Univariate example
data(oxtemp)
(fit = vglm(maxtemp ~ 1, egev, data=oxtemp, trace=TRUE))
fitted(fit)[1:3,]
coef(fit, mat=TRUE)
Coef(fit)
vcov(fit)
vcov(fit, untransform=TRUE)
sqrt(diag(vcov(fit)))   # Approximate standard errors
rlplot(fit)