morgenstern {VGAM} | R Documentation

## Morgenstern's Bivariate Distribution Family Function

### Description

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

### Usage

    morgenstern(lapar="rhobit", earg=list(), iapar=NULL, tola0=0.01,
                method.init=1)

### Arguments

`lapar` | Link function applied to the association parameter alpha, which lies between -1 and 1. 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`.
`tola0` | Positive numeric. If the estimate of alpha has an absolute value less than this then it is replaced by this value. This is an attempt to fix a numerical problem when the estimate is too close to zero.
`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`.

### Details

The cumulative distribution function is

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

for alpha between -1 and 1. The support of the function is for y1>0 and y2>0. The marginal distributions are an exponential distribution with unit mean. When alpha=0 then the random variables are independent, and this causes some problems in the estimation process since the distribution no longer depends on the parameter.

A variant of Newton-Raphson is used, which only seems to work for an intercept model. It is a very good idea to set `trace=TRUE`.

### 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 1. This is because each marginal distribution corresponds to a exponential distribution with unit mean.

This VGAM family function should be used with caution.

### 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

`fgm`, `gumbelIbiv`.

### Examples

    n = 1000
    ymat = cbind(rexp(n), rexp(n))
    ## Not run: plot(ymat)
    fit = vglm(ymat ~ 1, fam=morgenstern, trace=TRUE)
    fit = vglm(ymat ~ 1, fam=morgenstern, trace=TRUE, crit="coef")
    coef(fit, matrix=TRUE)
    Coef(fit)
    head(fitted(fit))

[Package VGAM version 0.7-9 Index]