Farlie-Gumbel-Morgenstern's Bivariate Distribution Family Function

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

Usage

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 $$P(Y_1 \leq y_1, Y_2 \leq y_2) = y_1 y_2 ( 1 + \alpha (1 - y_1) (1 - y_2) ) $$ 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.

References

Castillo, E., Hadi, A. S., Balakrishnan, N. and 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.

Author

T. W. Yee

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.

Examples

ymat <- rbifgmcop(1000, apar = rhobitlink(3, inverse = TRUE))
if (FALSE) plot(ymat, col = "blue") # \dontrun{}
fit <- vglm(ymat ~ 1, fam = bifgmcop, trace = TRUE)
#> Iteration 1: loglikelihood = 38.64911
#> Iteration 2: loglikelihood = 38.649281
#> Iteration 3: loglikelihood = 38.649282
#> Iteration 4: loglikelihood = 38.649282
coef(fit, matrix = TRUE)
#>             rhobitlink(apar)
#> (Intercept)         2.194155
Coef(fit)
#>      apar 
#> 0.7994468 
head(fitted(fit))
#>   [,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