frechet: Frechet Distribution Family Function

In VGAM: Vector Generalized Linear and Additive Models

Frechet Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Frechet distribution.

Usage

frechet(location = 0, lscale = "loglink",
  lshape = logofflink(offset = -2),
  iscale = NULL, ishape = NULL, nsimEIM = 250, zero = NULL)

Arguments

location	Numeric. Location parameter. It is called a below.
lscale, lshape	Link functions for the parameters; see Links for more choices.
iscale, ishape, zero, nsimEIM	See CommonVGAMffArguments for information.

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 frechet 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.

Value

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

Warning

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

Author(s)

T. W. Yee

References

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

See Also

rfrechet, gev.

Examples

## Not run: 
set.seed(123)
fdata <- data.frame(y1 = rfrechet(1000, shape = 2 + exp(1)))
with(fdata, hist(y1))
fit2 <- vglm(y1 ~ 1, frechet, data = fdata, trace = TRUE)
coef(fit2, matrix = TRUE)
Coef(fit2)
head(fitted(fit2))
with(fdata, mean(y1))
head(weights(fit2, type = "working"))
vcov(fit2)

## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.