# bifgmexp: Bivariate Farlie-Gumbel-Morgenstern Exponential Distribution... In VGAM: Vector Generalized Linear and Additive Models

## Description

Estimate the association parameter of FGM bivariate exponential distribution by maximum likelihood estimation.

## Usage

 `1` ```bifgmexp(lapar = "rhobitlink", iapar = NULL, tola0 = 0.01, imethod = 1) ```

## Arguments

 `lapar` Link function for the association parameter alpha, which lies between -1 and 1. See `Links` for more choices and other 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 `imethod`. `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. `imethod` 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.

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.

`bifgmcop`, `bigumbelIexp`.

## Examples

 ```1 2 3 4 5 6 7 8``` ```N <- 1000; mdata <- data.frame(y1 = rexp(N), y2 = rexp(N)) ## Not run: plot(ymat) fit <- vglm(cbind(y1, y2) ~ 1, bifgmexp, data = mdata, trace = TRUE) fit <- vglm(cbind(y1, y2) ~ 1, bifgmexp, data = mdata, # This may fail trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) head(fitted(fit)) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -2024.0218
VGLM    linear loop  2 :  loglikelihood = -2024.0328
Taking a modified step....................
Warning messages:
1: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
iterations terminated because half-step sizes are very small
2: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
VGLM    linear loop  1 :  coefficients = -0.012544349
VGLM    linear loop  2 :  coefficients = -0.028685601
VGLM    linear loop  3 :  coefficients = 0.0038774824
VGLM    linear loop  4 :  coefficients = -0.01226377
VGLM    linear loop  5 :  coefficients = -0.028405022
VGLM    linear loop  6 :  coefficients = 0.0038770692
VGLM    linear loop  7 :  coefficients = -0.012264183
VGLM    linear loop  8 :  coefficients = -0.028405435
VGLM    linear loop  9 :  coefficients = 0.0038770698
VGLM    linear loop  10 :  coefficients = -0.012264182
VGLM    linear loop  11 :  coefficients = -0.028405435
VGLM    linear loop  12 :  coefficients = 0.0038770698
VGLM    linear loop  13 :  coefficients = -0.012264182
VGLM    linear loop  14 :  coefficients = -0.028405435
VGLM    linear loop  15 :  coefficients = 0.0038770698
VGLM    linear loop  16 :  coefficients = -0.012264182
VGLM    linear loop  17 :  coefficients = -0.028405435
VGLM    linear loop  18 :  coefficients = 0.0038770698
VGLM    linear loop  19 :  coefficients = -0.012264182
VGLM    linear loop  20 :  coefficients = -0.028405435
VGLM    linear loop  21 :  coefficients = 0.0038770698
VGLM    linear loop  22 :  coefficients = -0.012264182
VGLM    linear loop  23 :  coefficients = -0.028405435
VGLM    linear loop  24 :  coefficients = 0.0038770698
VGLM    linear loop  25 :  coefficients = -0.012264182
VGLM    linear loop  26 :  coefficients = -0.028405435
VGLM    linear loop  27 :  coefficients = 0.0038770698
VGLM    linear loop  28 :  coefficients = -0.012264182
VGLM    linear loop  29 :  coefficients = -0.028405435
VGLM    linear loop  30 :  coefficients = 0.0038770698
Warning message:
In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
convergence not obtained in 30 IRLS iterations
rhobit(apar)
(Intercept)   0.00387707
apar
0.001938532
y1 y2
1  1  1
2  1  1
3  1  1
4  1  1
5  1  1
6  1  1
```

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