Ali-Mikhail-Haq Distribution Family Function

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

Usage

biamhcop(lapar = "rhobitlink", iapar = NULL, imethod = 1,
         nsimEIM = 250)

Arguments

lapar

Link function applied to the association parameter \(\alpha\), which is real and \(-1 < \alpha < 1\). See Links for more choices.

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.

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 iapar.

nsimEIM

See CommonVGAMffArguments for more information.

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. This is an Archimedean copula.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

References

Balakrishnan, N. and Lai, C.-D. (2009). Continuous Bivariate Distributions, 2nd ed. New York: Springer.

Author

T. W. Yee and C. S. Chee

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.

See also

rbiamhcop, bifgmcop, bigumbelIexp, rbilogis, simulate.vlm.

Examples

ymat <- rbiamhcop(1000, apar = rhobitlink(2, inverse = TRUE))
fit <- vglm(ymat ~ 1, biamhcop, trace = TRUE)
#> Iteration 1: loglikelihood = 51.436617
#> Iteration 2: loglikelihood = 51.438348
#> Iteration 3: loglikelihood = 51.438356
#> Iteration 4: loglikelihood = 51.438356
coef(fit, matrix = TRUE)
#>             rhobitlink(apar)
#> (Intercept)         1.850311
Coef(fit)
#>      apar 
#> 0.7283273