VGAM (version 0.9-4)

gumbelII: Gumbel-II Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Gumbel-II distribution.

Usage

gumbelII(lshape = "loge", lscale = "loge",
         ishape = NULL,   iscale = NULL,
         probs.y = c(0.2, 0.5, 0.8),
         perc.out = NULL, imethod = 1, zero = -2)

Arguments

lshape, lscale

Parameter link functions applied to the (positive) shape parameter (called $a$ below) and (positive) scale parameter (called $b$ below). See Links for more choices.

ishape, iscale

Optional initial values for the shape and scale parameters.

imethod

See weibull.

zero, probs.y

Details at CommonVGAMffArguments.

perc.out

If the fitted values are to be quantiles then set this argument to be the percentiles of these, e.g., 50 for median.

Value

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

Details

The Gumbel-II density for a response $Y$ is $$f(y;a,b) = a y^{a-1} \exp[-(y/b)^a] / (b^a)$$ for $a > 0$, $b > 0$, $y > 0$. The cumulative distribution function is $$F(y;a,b) = \exp[-(y/b)^{-a}].$$ The mean of $Y$ is $b \, \Gamma(1 - 1/a)$ (returned as the fitted values) when $a>1$, and the variance is $b^2\,\Gamma(1-2/a)$ when $a>2$. This distribution looks similar to weibull, and is due to Gumbel (1954).

This VGAM family function currently does not handle censored data. Fisher scoring is used to estimate the two parameters. Probably similar regularity conditions hold for this distribution compared to the Weibull distribution.

References

Gumbel, E. J. (1954). Statistical theory of extreme values and some practical applications. Applied Mathematics Series, volume 33, U.S. Department of Commerce, National Bureau of Standards, USA.

See Also

dgumbelII, gumbel, gev.

Examples

gdata <- data.frame(x2 = runif(nn <- 1000))
gdata <- transform(gdata, eta1  = +1,
                          eta2  = -1 + 0.1 * x2,
                          ceta1 =  0,
                          ceta2 =  1)
gdata <- transform(gdata, shape1 = exp(eta1),
                          shape2 = exp(eta2),
                          scale1 = exp(ceta1),
                          scale2 = exp(ceta2))
gdata <- transform(gdata,
                   y1 = rgumbelII(nn, shape = shape1, scale = scale1),
                   y2 = rgumbelII(nn, shape = shape2, scale = scale2))

fit <- vglm(cbind(y1, y2) ~ x2,
            gumbelII(zero = c(1, 2, 4)), data = gdata, trace = TRUE)
coef(fit, matrix = TRUE)
vcov(fit)
summary(fit)