amh {VGAM} | R Documentation

## Ali-Mikhail-Haq Distribution Distribution Family Function

### Description

Estimate the association parameter of Ali-Mikhail-Haq's bivariate distribution by maximum likelihood estimation.

### Usage

    amh(lalpha="rhobit", ealpha=list(), ialpha=NULL,
        method.init=1, nsimEIM=250)

### Arguments

`lalpha` | Link function applied to the association parameter alpha, which is real and -1 < alpha < 1. See `Links` for more choices.
`ealpha` | List. Extra argument for the link. See `earg` in `Links` for general information.
`ialpha` | 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 `ialpha`.
`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 and C. S. Chee

### References

Hutchinson, T. P. and Lai, C. D. (1990) Continuous Bivariate Distributions, Emphasising Applications, Adelaide, South Australia: Rumsby Scientific Publishing.

### See Also

`ramh`, `fgm`, `gumbelIbiv`.

### Examples

    ymat = ramh(1000, alpha=rhobit(2, inverse=TRUE))
    fit = vglm(ymat ~ 1, amh, trace = TRUE)
    coef(fit, mat=TRUE)
    Coef(fit)

[Package VGAM version 0.7-9 Index]