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),
    ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
    gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
    gshape = (-5:5) / 11 + 0.01,
    iprobs.y = NULL, tolshape0 = 0.001,
    type.fitted = c("percentiles", "mean"),
    zero = c("scale", "shape"))
gevff(llocation = "identitylink", lscale = "loge",
    lshape = logoff(offset = 0.5), percentiles = c(95, 99),
    ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
    gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
    gshape = (-5:5) / 11 + 0.01,
    iprobs.y = NULL, tolshape0 = 0.001,
    type.fitted = c("percentiles", "mean"), 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 an offset called $A$ below; and then the linear/additive predictor is $\log(\xi+A)$ which means that $\xi > -A$. For technical reasons (see Details) it is a good idea for $A = 0.5$.

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 $\mu + \sigma (\Gamma(1-\xi)-1) / \xi$ is returned as the fitted values, and these are only defined for $\xi<1$.

ilocation, iscale, ishape

Numeric. Initial value for the location parameter, $\sigma$ and $\xi$. A NULL means a value is computed internally. The argument ishape is more important than the other two. If a failure to converge occurs, or even to obtain initial values occurs, try assigning ishape some value (positive or negative; the sign can be very important). Also, in general, a larger value of iscale tends to be better than a smaller value.

imethod

Initialization method. Either the value 1 or 2. If both methods fail then try using ishape. See CommonVGAMffArguments for information.

gshape

Numeric vector. The values are used for a grid search for an initial value for $\xi$. See CommonVGAMffArguments for information.

gprobs.y, gscale.mux, iprobs.y

Numeric vectors, used for the initial values. See CommonVGAMffArguments for information.

tolshape0

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 modelled as a linear combination of the explanatory variables. For many data sets having zero = 3 is a good idea. See CommonVGAMffArguments for information.

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 may occur for gev() with multivariate responses. In general, gevff() 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, reverse Weibull, and Gumbel types respectively. It can be noted that the Gumbel (or Type I) distribution accommodates many commonly-used distributions such as the normal, lognormal, logistic, gamma, exponential and Weibull.

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.

Examples

Run this code

# NOT RUN {
# 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, gevff, 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)
# }

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