fgm: Farlie-Gumbel-Morgenstern's Bivariate Distribution Family Function

Description

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

Usage

fgm(lapar = "rhobit", iapar = NULL, imethod = 1)

Arguments

lapar, iapar, imethod

Details at CommonVGAMffArguments. See Links for more link function choices.

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 + α (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

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.

Smith, M. D. (2007) Invariance theorems for Fisher information. Communications in Statistics---Theory and Methods, 36(12), 2213--2222.

Examples

Run this code

ymat <- rfgm(n = 1000, alpha = rhobit(3, inverse = TRUE))
plot(ymat, col = "blue")
fit <- vglm(ymat ~ 1, fam = fgm, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))

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