Bivariate Farlie-Gumbel-Morgenstern Exponential Distribution Family Function

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

Usage

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 $$P(Y_1 \leq y_1, Y_2 \leq y_2) = e^{-y_1-y_2} ( 1 + \alpha [1 - e^{-y_1}] [1 - e^{-y_2}] ) + 1 - e^{-y_1} - e^{-y_2} $$ for $\alpha$ between $-1$ and $1$. The support of the function is for $y_1>0$ and $y_2>0$. 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.

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.

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

This VGAM family function should be used with caution.

Examples

N <- 1000; mdata <- data.frame(y1 = rexp(N), y2 = rexp(N))
if (FALSE) plot(ymat) # \dontrun{}
fit <- vglm(cbind(y1, y2) ~ 1, bifgmexp, data = mdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1994.6345
#> Iteration 2: loglikelihood = -1994.554
#> Iteration 3: loglikelihood = -1994.554
#> Iteration 4: loglikelihood = -1994.554
fit <- vglm(cbind(y1, y2) ~ 1, bifgmexp, data = mdata, # May fail
            trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = -0.23182787
#> Iteration 2: coefficients = -0.15545762
#> Iteration 3: coefficients = -0.15628525
#> Iteration 4: coefficients = -0.15628532
#> Iteration 5: coefficients = -0.15628532
coef(fit, matrix = TRUE)
#>             rhobitlink(apar)
#> (Intercept)       -0.1562853
Coef(fit)
#>      apar 
#> -0.077984 
head(fitted(fit))
#>   y1 y2
#> 1  1  1
#> 2  1  1
#> 3  1  1
#> 4  1  1
#> 5  1  1
#> 6  1  1