VGAM (version 0.7-1)

frechet: Frechet Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter and 3-parameter Frechet distribution.

Usage

frechet2(location=0, lscale="loge", lshape="loglog",
         escale = list(), eshape = list(),
         iscale=NULL, ishape=3, zero=NULL)
frechet3(anchor=NULL, ldifference="loge",
         lscale="loge", lshape="loglog",
         edifference=list(), escale = list(), eshape = list(),
         ilocation=NULL, iscale=NULL, ishape=3, zero=NULL,
         effpos = .Machine$double.eps^0.75)

Value

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

Warning

Convergence for frechet3 can be very slow, especially if the initial value for the location parameter is poor. Setting something like maxit=200, trace=TRUE is a good idea.

Details

The 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). Note that the loglog link ensures $s>1$. The variance of $Y$ is $b^2 [ \Gamma(1-2/s) - \Gamma^2(1-1/s)]$ for $s>2$.

frechet2 has $a$ known whereas frechet3 estimates it. Estimating $a$ well requires a lot of data and a good choice of ilocation will help speed up convergence. For frechet3 the default linear/additive predictors are $\log(D)$, $\log(b)$, $\log\log(s)$, respectively. It would be great if the first linear/additive predictor was a direct function of the location parameter, but this can run the risk that the estimate is out of range (i.e., greater than min(y)).

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.

Examples

y = rfrechet(n <- 1000, shape=exp(exp(0)))
hist(y)
fit2 = vglm(y ~ 1, frechet2, trace=TRUE, maxit=155)
fit3 = vglm(y ~ 1, frechet3(ilocation=0), trace=TRUE, maxit=155)
coef(fit3, matrix=TRUE)
Coef(fit3)
fitted(fit3)[1:5,]
mean(y)
weights(fit3, type="w")[1:5,]
vcov(fit3)   # caution needed!
fit3@extra$location[1:3]  # Estimate of the location parameter
fit3@extra$LHSanchor  # Anchor point
min(y)