gumbel {VGAM} | R Documentation

## Gumbel Distribution Family Function

### Description

Maximum likelihood estimation of the 2-parameter Gumbel distribution.

### Usage

    gumbel(llocation = "identitylink", lscale = "loglink",
           iscale = NULL, R = NA, percentiles = c(95, 99),
           mpv = FALSE, zero = NULL)
    gumbelff(llocation = "identitylink", lscale = "loglink",
             iscale = NULL, R = NA, percentiles = c(95, 99),
             zero = "scale", mpv = FALSE)

### Arguments

`llocation, lscale` | Parameter link functions for mu and sigma. See `Links` for more choices.
`iscale` | Numeric and positive. Optional initial value for sigma. Recycled to the appropriate length. In general, a larger value is better than a smaller value. A `NULL` means an initial value is computed internally.
`R` | Numeric. Maximum number of values possible. See Details for more details.
`percentiles` | Numeric vector of percentiles used for the fitted values. Values should be between 0 and 100. This argument uses the argument `R` if assigned. If `percentiles = NULL` then the mean will be returned as the fitted values.
`mpv` | Logical. If `mpv = TRUE` then the median predicted value (MPV) is computed and returned as the (last) column of the fitted values. This argument is ignored if `percentiles = NULL`. See Details for more details.

`zero` | A vector specifying which linear/additive predictors are modelled as intercepts only. The value (possibly values) can be from the set {1, 2} corresponding respectively to mu and sigma. By default all linear/additive predictors are modelled as a linear combination of the explanatory variables. See `CommonVGAMffArguments` for more details.

### Details

The Gumbel distribution is a generalized extreme value (GEV) distribution with shape parameter xi = 0. Consequently it is more easily estimated than the GEV. See `gev` for more details.

The quantity R is the maximum number of observations possible, for example, in the Venice data below, the top 10 daily values are recorded for each year, therefore R = 365 because there are about 365 days per year. The MPV is the value of the response such that the probability of obtaining a value greater than the MPV is 0.5 out of R observations. For the Venice data, the MPV is the sea level such that there is an even chance that the highest level for a particular year exceeds the MPV. If `mpv = TRUE` then the column labelled `"MPV"` contains the MPVs when `fitted()` is applied to the fitted object.

The formula for the mean of a response Y is μ+σ \times Euler where Euler is a constant that has value approximately equal to 0.5772. The formula for the percentiles are (if `R` is not given) location- scale*log[-log(P/100)] where P is the `percentile` argument value(s). If `R` is given then the percentiles are location- scale*log[-log(R*(1-P/100))].

### Value

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

### Warning

When `R` is not given (the default) the fitted percentiles are that of the data, and not of the overall population. For example, in the example below, the 50 percentile is approximately the running median through the data, however, the data are the highest sea level measurements recorded each year (it therefore equates to the median predicted value or MPV).

### Note

Like many other usual VGAM family functions, `gumbelff()` handles (independent) multiple responses.

`gumbel()` can handle more of a multivariate response, i.e., a matrix with more than one column. Each row of the matrix is sorted into descending order. Missing values in the response are allowed but require `na.action = na.pass`. The response matrix needs to be padded with any missing values. With a multivariate response one has a matrix `y`, say, where `y[, 2]` contains the second order statistics, etc.

### 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.

Smith, R. L. (1986) Extreme value theory based on the r largest annual events. Journal of Hydrology, 86, 27–43.

Rosen, O. and Cohen, A. (1996) Extreme percentile regression. In: Haerdle, W. and Schimek, M. G. (eds.), Statistical Theory and Computational Aspects of Smoothing: Proceedings of the COMPSTAT '94 Satellite Meeting held in Semmering, Austria, 27–28 August 1994, pp.200–214, Heidelberg: Physica-Verlag.

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

### See Also

`rgumbel`, `dgumbelII`, `cens.gumbel`, `guplot`, `gev`, `gevff`, `venice`.

### Examples

    # Example 1: Simulated data
    gdata <- data.frame(y1 = rgumbel(n = 1000, loc = 100, scale = exp(1)))
    fit1 <- vglm(y1 ~ 1, gumbelff(perc = NULL), data = gdata, trace = TRUE)
    coef(fit1, matrix = TRUE)
    Coef(fit1)
    head(fitted(fit1))
    with(gdata, mean(y1))

    # Example 2: Venice data
    (fit2 <- vglm(cbind(r1, r2, r3, r4, r5) ~ year, data = venice,
                  gumbel(R = 365, mpv = TRUE), trace = TRUE))
    head(fitted(fit2))
    coef(fit2, matrix = TRUE)
    sqrt(diag(vcov(summary(fit2))))   # Standard errors

    # Example 3: Try a nonparametric fit ---------------------
    # Use the entire data set, including missing values
    # Same as as.matrix(venice[, paste0("r", 1:10)]):
    Y <- Select(venice, "r", sort = FALSE)
    fit3 <- vgam(Y ~ s(year, df = 3), gumbel(R = 365, mpv = TRUE),
                 data = venice, trace = TRUE, na.action = na.pass)
    depvar(fit3)[4:5, ]  # NAs used to pad the matrix

    ## Not run:   # Plot the component functions
    par(mfrow = c(2, 3), mar = c(6, 4, 1, 2) + 0.3, xpd = TRUE)
    plot(fit3, se = TRUE, lcol = "blue", scol = "limegreen", lty = 1,
         lwd = 2, slwd = 2, slty = "dashed")

    # Quantile plot --- plots all the fitted values
    qtplot(fit3, mpv = TRUE, lcol = c(1, 2, 5), tcol = c(1, 2, 5), lwd = 2,
           pcol = "blue", tadj = 0.1, ylab = "Sea level (cm)")

    # Plot the 99 percentile only
    year <- venice[["year"]]
    matplot(year, Y, ylab = "Sea level (cm)", type = "n")
    matpoints(year, Y, pch = "*", col = "blue")
    lines(year, fitted(fit3)[, "99%"], lwd = 2, col = "orange")

    # Check the 99 percentiles with a smoothing spline.
    # Nb. (1-0.99) * 365 = 3.65 is approx. 4, meaning the 4th order
    # statistic is approximately the 99 percentile.
    plot(year, Y[, 4], ylab = "Sea level (cm)", type = "n",
         main = "Orange is 99 percentile, Green is a smoothing spline")
    points(year, Y[, 4], pch = "4", col = "blue")
    lines(year, fitted(fit3)[, "99%"], lty = 1, col = "orange")
    lines(smooth.spline(year, Y[, 4], df = 4), col = "limegreen", lty = 2)

    ## End(Not run)

[Package VGAM version 1.1-1 Index]