VGAM (version 0.8-4.1)

frechet: Frechet Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Frechet distribution.

Usage

frechet2(location = 0, lscale = "loge", lshape = "logoff",
         escale = list(), eshape = list(offset = -2), iscale = NULL,
         ishape = NULL, nsimEIM = 250, zero = NULL)

Arguments

location

Numeric. Location parameter. It is called $a$ below.

lscale, lshape, escale, eshape

Link functions and extra arguments for the parameters. See Links for more choices.

iscale, ishape, zero, nsimEIM

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

Family function frechet2 may fail for low values of the shape parameter, e.g., near 2 or lower.

Details

The (3-parameter) Frechet distribution has a density function that can be written $$f(y) = \frac{sb}{(y-a)^2} [b/(y-a)]^{s-1} \, \exp[-(b/(y-a))^s]$$ for $y > a$ and scale parameter $b > 0$. The positive shape parameter is $s$. The cumulative distribution function is $$F(y) = \exp[-(b/(y-a))^s].$$ The mean of $Y$ is $a + b \Gamma(1-1/s)$ for $s > 1$ (these are returned as the fitted values). The variance of $Y$ is $b^2 [ \Gamma(1-2/s) - \Gamma^2(1-1/s)]$ for $s > 2$.

Family frechet2 has $a$ known, and $\log(b)$ and $\log(s - 2)$ are the default linear/additive predictors. The working weights are estimated by simulated Fisher scoring.

References

Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005) Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, N.J.: Wiley-Interscience.

See Also

rfrechet, gev.

Examples

set.seed(123)
fdata = data.frame(y1 = rfrechet(nn <- 1000, shape = 2 + exp(1)))
with(fdata, hist(y1))
fit2 = vglm(y1 ~ 1, frechet2, fdata, trace = TRUE)
coef(fit2, matrix = TRUE)
Coef(fit2)
head(fitted(fit2))
with(fdata, mean(y1))
head(weights(fit2, type = "working"))
vcov(fit2)