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.
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.
Details
The Benini distribution
has a probability density function that can be written
$$f(y) = 2 b \exp(-b[(\log(y/y_0))^2]) \log(y/y_0) / y$$
for $0 < y_0 < y$, and $b>0$.
The cumulative distribution function for $Y$ is
$$F(y) = 1 - \exp(-b[(\log(y/y_0))^2]).$$
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.
References
Kleiber, C. and Kotz, S. (2003)
Statistical Size Distributions in Economics and
Actuarial Sciences,
Hoboken, NJ: Wiley-Interscience.