benini {VGAM}R Documentation

Benini Distribution Family Function

Description

Estimating the parameter of the Benini distribution by maximum likelihood estimation.

Usage

benini(y0=stop("argument \"y0\" must be specified"),
       lshape="loge", earg=list(), ishape=NULL, method.init=1)

Arguments

y0 Positive scale parameter.
lshape Parameter link function applied to the parameter b, which is the shape parameter. See Links for more choices. A log link is the default because b is positive.
earg List. Extra argument for the link. See earg in Links for general information.
ishape Optional initial value for the shape parameter. The default is to compute the value internally.
method.init An integer with value 1 or 2 which specifies the initialization method. If failure to converge occurs try the other value, or else specify a value for ishape.

Details

The Benini distribution has a probability density function that can be written

f(y) = 2*b*exp(-b * [(log(y/y0))^2]) * log(y/y0) / y

for y_0>0, y0<y, and b>0. The cumulative distribution function for Y is

F(y) = 1 - exp(-b * [(log(y/y0))^2]).

Here, Newton-Raphson and Fisher scoring coincide.

On fitting, the extra slot has a component called y0 which contains the value of the y0 argument.

Value

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

Warning

The mean of Y, which are returned as the fitted values, may be incorrect.

Note

Yet to do: the 2-parameter Benini distribution estimates y0 as well, and the 3-parameter Benini distribution estimates another shape parameter a too.

Author(s)

T. W. Yee

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.

See Also

Benini.

Examples

y  = rbenini(n <- 3000, y0=1, shape=exp(2))
fit = vglm(y ~ 1, benini(y0=1), trace=TRUE, crit="c")
coef(fit, matrix=TRUE)
Coef(fit)
fit@extra$y0

# Apparent discrepancy:
fitted(fit)[1:5]
mean(y)

[Package VGAM version 0.7-7 Index]