fgm {VGAM} | R Documentation

## Farlie-Gumbel-Morgenstern's Bivariate Distribution Family Function

### Description

Estimate the association parameter of Farlie-Gumbel-Morgenstern's bivariate distribution by maximum likelihood estimation.

### Usage

    fgm(lapar="rhobit", earg=list(), iapar=NULL,
        method.init=1, nsimEIM=200)

### Arguments

`lapar` | Link function applied to the association parameter alpha, which is real. See `Links` for more choices.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`iapar` | Numeric. Optional initial value for alpha. By default, an initial value is chosen internally. If a convergence failure occurs try assigning a different value. Assigning a value will override the argument `method.init`.
`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 `ia`.
`nsimEIM` | See `CommonVGAMffArguments` for more information.

### Details

The cumulative distribution function is

P(Y1 <= y1, Y2 <= y2) = y1 * y2 * ( 1 + alpha * (1 - y1) * (1 - y2) )

for -1 < alpha < 1. The support of the function is the unit square. The marginal distributions are the standard uniform distributions. When alpha=0 the random variables are independent.

### Value

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

### Note

The response must be a two-column matrix. Currently, the fitted value is a matrix with two columns and values equal to 0.5. This is because each marginal distribution corresponds to a standard uniform distribution.

### Author(s)

T. W. Yee

### 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

`rfgm`, `frank`, `morgenstern`.

### Examples

    ymat = rfgm(n = 1000, alpha=rhobit(3, inverse=TRUE))
    ## Not run: plot(ymat, col="blue")
    fit = vglm(ymat ~ 1, fam=fgm, trace=TRUE)
    coef(fit, matrix=TRUE)
    Coef(fit)
    head(fitted(fit))

[Package VGAM version 0.7-9 Index]