amh: Ali-Mikhail-Haq Distribution Distribution Family Function

Description

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

Usage

amh(lalpha = "rhobit", ealpha = list(), ialpha = NULL,
    imethod = 1, nsimEIM = 250)

Arguments

lalpha

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

ealpha

List. Extra argument for the link. See earg in Links for general information.

ialpha

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 ialpha.

nsimEIM

See CommonVGAMffArguments for more information.

Value

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

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.

References

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

Examples

Run this code

ymat <- ramh(1000, alpha = rhobit(2, inverse = TRUE))
fit <- vglm(ymat ~ 1, amh, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)

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