VGAM (version 1.1-12)

biclaytoncop: Clayton Copula (Bivariate) Family Function

Description

Estimate the correlation parameter of the (bivariate) Clayton copula distribution by maximum likelihood estimation.

Usage

biclaytoncop(lapar = "loglink", iapar = NULL, imethod = 1,
             parallel = FALSE, zero = NULL)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Arguments

lapar, iapar, imethod

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

parallel, zero

Details at CommonVGAMffArguments. If parallel = TRUE then the constraint is also applied to the intercept.

Author

R. Feyter and T. W. Yee

Details

The cumulative distribution function is $$P(u_1, u_2;\alpha) = (u_1^{-\alpha} + u_2^{-\alpha}-1)^{-1/\alpha}$$ for \(0 \leq \alpha \). Here, \(\alpha\) is the association parameter. The support of the function is the interior of the unit square; however, values of 0 and/or 1 are not allowed (currently). The marginal distributions are the standard uniform distributions. When \(\alpha = 0\) the random variables are independent.

This VGAM family function can handle multiple responses, for example, a six-column matrix where the first 2 columns is the first out of three responses, the next 2 columns being the next response, etc.

References

Clayton, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society, Series B, Methodological, 44, 414--422.

Schepsmeier, U. and Stober, J. (2014). Derivatives and Fisher information of bivariate copulas. Statistical Papers 55, 525--542.

See Also

rbiclaytoncop, dbiclaytoncop, kendall.tau.

Examples

ymat <- rbiclaytoncop(n = (nn <- 1000), apar = exp(2))
bdata <- data.frame(y1 = ymat[, 1], y2 = ymat[, 2],
                    y3 = ymat[, 1], y4 = ymat[, 2], x2 = runif(nn))
summary(bdata)
if (FALSE)  plot(ymat, col = "blue") 
fit1 <-
  vglm(cbind(y1, y2, y3, y4) ~ 1,  # 2 responses, e.g., (y1,y2) is the 1st
       biclaytoncop, data = bdata,
       trace = TRUE, crit = "coef")  # Sometimes a good idea
coef(fit1, matrix = TRUE)
Coef(fit1)
head(fitted(fit1))
summary(fit1)

# Another example; apar is a function of x2
bdata <- transform(bdata, apar = exp(-0.5 + x2))
ymat <- rbiclaytoncop(n = nn, apar = with(bdata, apar))
bdata <- transform(bdata, y5 = ymat[, 1], y6 = ymat[, 2])
fit2 <- vgam(cbind(y5, y6) ~ s(x2), data = bdata,
             biclaytoncop(lapar = "loglink"), trace = TRUE)
if (FALSE) plot(fit2, lcol = "blue", scol = "orange", se = TRUE)