# bifgmcop: Farlie-Gumbel-Morgenstern's Bivariate Distribution Family... In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 `1` ```bifgmcop(lapar = "rhobitlink", iapar = NULL, imethod = 1) ```

## Arguments

 `lapar, iapar, imethod` Details at `CommonVGAMffArguments`. See `Links` for more link function choices.

## 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.

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, NJ, USA: Wiley-Interscience.

Smith, M. D. (2007). Invariance theorems for Fisher information. Communications in Statistics—Theory and Methods, 36(12), 2213–2222.

`rbifgmcop`, `bifrankcop`, `bifgmexp`, `simulate.vlm`.

## Examples

 ```1 2 3 4 5 6``` ```ymat <- rbifgmcop(n = 1000, apar = rhobitlink(3, inverse = TRUE)) ## Not run: plot(ymat, col = "blue") fit <- vglm(ymat ~ 1, fam = bifgmcop, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) head(fitted(fit)) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = 41.676716
VGLM    linear loop  2 :  loglikelihood = 41.676821
VGLM    linear loop  3 :  loglikelihood = 41.676821
VGLM    linear loop  4 :  loglikelihood = 41.676821
(Intercept)         2.502279
apar
0.8486028
[,1] [,2]
1  0.5  0.5
2  0.5  0.5
3  0.5  0.5
4  0.5  0.5
5  0.5  0.5
6  0.5  0.5
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.