gumbelII.RdMaximum likelihood estimation of the 2-parameter Gumbel-II distribution.
gumbelII(lscale = "loglink", lshape = "loglink", iscale = NULL, ishape = NULL,
probs.y = c(0.2, 0.5, 0.8), perc.out = NULL, imethod = 1,
zero = "shape", nowarning = FALSE)Logical. Suppress a warning?
Parameter link functions applied to the
(positive) shape parameter (called \(s\) below) and
(positive) scale parameter (called \(b\) below).
See Links for more choices.
Parameter link functions applied to the
Optional initial values for the shape and scale parameters.
See weibullR.
Details at CommonVGAMffArguments.
If the fitted values are to be quantiles then set this argument to be the percentiles of these, e.g., 50 for median.
The Gumbel-II density for a response \(Y\) is
$$f(y;b,s) = s y^{s-1} \exp[-(y/b)^s] / (b^s)$$
for \(b > 0\), \(s > 0\), \(y > 0\).
The cumulative distribution function is
$$F(y;b,s) = \exp[-(y/b)^{-s}].$$
The mean of \(Y\) is \(b \, \Gamma(1 - 1/s)\)
(returned as the fitted values)
when \(s>1\),
and the variance is \(b^2\,\Gamma(1-2/s)\) when
\(s>2\).
This distribution looks similar to weibullR, 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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
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 weibullR.
This VGAM family function handles multiple responses.
gdata <- data.frame(x2 = runif(nn <- 1000))
gdata <- transform(gdata, heta1 = +1,
heta2 = -1 + 0.1 * x2,
ceta1 = 0,
ceta2 = 1)
gdata <- transform(gdata, shape1 = exp(heta1),
shape2 = exp(heta2),
scale1 = exp(ceta1),
scale2 = exp(ceta2))
gdata <- transform(gdata,
y1 = rgumbelII(nn, scale = scale1, shape = shape1),
y2 = rgumbelII(nn, scale = scale2, shape = shape2))
fit <- vglm(cbind(y1, y2) ~ x2,
gumbelII(zero = c(1, 2, 3)), data = gdata, trace = TRUE)
#> Iteration 1: loglikelihood = -5992.6654
#> Iteration 2: loglikelihood = -5992.6631
#> Iteration 3: loglikelihood = -5992.6631
coef(fit, matrix = TRUE)
#> loglink(scale1) loglink(shape1) loglink(scale2) loglink(shape2)
#> (Intercept) -0.00349692 1.024985 1.115326 -0.984855721
#> x2 0.00000000 0.000000 0.000000 -0.009192527
vcov(fit)
#> (Intercept):1 (Intercept):2 (Intercept):3 (Intercept):4
#> (Intercept):1 1.427292e-04 -9.222032e-05 0.000000e+00 0.0000000000
#> (Intercept):2 -9.222032e-05 6.079271e-04 0.000000e+00 0.0000000000
#> (Intercept):3 0.000000e+00 0.000000e+00 8.019324e-03 -0.0006943614
#> (Intercept):4 0.000000e+00 0.000000e+00 -6.943614e-04 0.0021854962
#> x2 0.000000e+00 0.000000e+00 6.347477e-06 -0.0032218031
#> x2
#> (Intercept):1 0.000000e+00
#> (Intercept):2 0.000000e+00
#> (Intercept):3 6.347477e-06
#> (Intercept):4 -3.221803e-03
#> x2 6.579751e-03
summary(fit)
#>
#> Call:
#> vglm(formula = cbind(y1, y2) ~ x2, family = gumbelII(zero = c(1,
#> 2, 3)), data = gdata, trace = TRUE)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept):1 -0.003497 0.011947 -0.293 0.77
#> (Intercept):2 1.024985 0.024656 41.571 <2e-16 ***
#> (Intercept):3 1.115326 0.089551 12.455 <2e-16 ***
#> (Intercept):4 -0.984856 0.046749 -21.067 <2e-16 ***
#> x2 -0.009193 0.081116 -0.113 0.91
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Names of linear predictors: loglink(scale1), loglink(shape1), loglink(scale2),
#> loglink(shape2)
#>
#> Log-likelihood: -5992.663 on 3995 degrees of freedom
#>
#> Number of Fisher scoring iterations: 3
#>
#> No Hauck-Donner effect found in any of the estimates
#>