benini1 {VGAM} | R Documentation

## Benini Distribution Family Function

### Description

Estimating the 1-parameter Benini distribution by maximum likelihood estimation.

### Usage

    benini1(y0 = stop("argument 'y0' must be specified"), lshape = "loglink",
            ishape = NULL, imethod = 1, zero = NULL, parallel = FALSE,
            type.fitted = c("percentiles", "Qlink"),
            percentiles = 50)

### Arguments

`y0` | Positive scale parameter.
`lshape` | Parameter link function and extra argument of the parameter b, which is the shape parameter. See `Links` for more choices. A log link is the default because b is positive.
`ishape` | Optional initial value for the shape parameter. The default is to compute the value internally.
`imethod, zero, parallel` | Details at `CommonVGAMffArguments`.
`type.fitted, percentiles` | See `CommonVGAMffArguments` for information. Using `"Qlink"` is for quantile-links in VGAMextra.

### Details

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

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

for 0 < y0 < y, and shape s > 0. The cumulative distribution function for Y is

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

Here, Newton-Raphson and Fisher scoring coincide. The median of Y is now returned as the fitted values, by default. This VGAM family function can handle a multiple responses, which is inputted as a matrix.

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`, and `vgam`.

### Note

Yet to do: the 2-parameter Benini distribution estimates another shape parameter a too. Hence, the code may change in the future.

### Author(s)

T. W. Yee

### References

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

### See Also

`Benini`.

### Examples

    y0 <- 1; nn <- 3000
    bdata <- data.frame(y  = rbenini(nn, y0 = y0, shape = exp(2)))
    fit <- vglm(y ~ 1, benini1(y0 = y0), data = bdata, trace = TRUE)
    coef(fit, matrix = TRUE)
    Coef(fit)
    fit@extra$y0
    c(head(fitted(fit), 1), with(bdata, median(y)))  # Should be equal

[Package VGAM version 1.1-1 Index]