gev {VGAM} | R Documentation

## 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.

### Details

The GEV distribution function can be written

G(y) = exp( -[ (y- mu)/ sigma ]_{+}^{- 1/ xi})

where sigma > 0, -Inf < mu < Inf, 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 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 kth 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.

### 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 <= 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.)

### Note

The VGAM family function `gev` can handle a multivariate (matrix) response, cf. multiple responses. If so, each row of the matrix is sorted into descending order and `NA`s are put last. With a vector or one-column matrix response using `gevff` will give the same result but be faster and it handles the xi = 0 case. The function `gev` implements Tawn (1988) while `gevff` implements Prescott and Walden (1980).

Function `egev()` has been replaced by the new family function `gevff()`. It now conforms to the usual VGAM philosophy of having `M1` linear predictors per (independent) response. This is the usual way multiple responses are handled. Hence `vglm(cbind(y1, y2)..., gevff, ...)` will have 6 linear predictors and it is possible to constrain the linear predictors so that the answer is similar to `gev()`. Missing values in the response of `gevff()` will be deleted; this behaviour is the same as with almost every other VGAM family function.

The shape parameter xi is difficult to estimate accurately unless there is a lot of data. Convergence is slow when xi is near -0.5. Given many explanatory variables, it is often a good idea to make sure `zero = 3`. The range restrictions of the parameter xi are not enforced; thus it is possible for a violation to occur.

Successful convergence often depends on having a reasonably good initial value for xi. If failure occurs try various values for the argument `ishape`, and if there are covariates, having `zero = 3` is advised.

### Author(s)

T. W. Yee

### 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`, `gumbelff`, `guplot`, `rlplot.gevff`, `gpd`, `weibullR`, `frechet`, `extlogit`, `oxtemp`, `venice`, `CommonVGAMffArguments`.

### Examples

    ## 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)

    ## End(Not run)

[Package VGAM version 1.0-3 Index]