binormalcop {VGAM}

Gaussian Copula (Bivariate) Family Function

Description

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

Usage

binormalcop(lrho = "rhobitlink", irho = NULL, imethod = 1,
            parallel = FALSE, zero = NULL)

Arguments

lrho, irho, imethod

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

parallel, zero

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

Details

The cumulative distribution function is

P(Y_1 \leq y_1, Y_2 \leq y_2) = \Phi_2 ( \Phi^{-1}(y_1), \Phi^{-1}(y_2); \rho )

for -1 < \rho < 1, \Phi_2 is the cumulative distribution function of a standard bivariate normal (see pbinorm), and \Phi is the cumulative distribution function of a standard univariate normal (see pnorm).

The support of the function is the interior of the unit square; however, values of 0 and/or 1 are not allowed. The marginal distributions are the standard uniform distributions. When \rho = 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.

Value

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

Note

The response matrix must have a multiple of two-columns. Currently, the fitted value is a matrix with the same number of columns and values equal to 0.5. This is because each marginal distribution corresponds to a standard uniform distribution.

This VGAM family function is fragile; each response must be in the interior of the unit square. Setting crit = "coef" is sometimes a good idea because inaccuracies in pbinorm might mean unnecessary half-stepping will occur near the solution.

Author(s)

T. W. Yee

References

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

See Also

rbinormcop, pnorm, kendall.tau.

Examples

nn <- 1000
ymat <- rbinormcop(nn, rho = rhobitlink(-0.9, inverse = TRUE))
bdata <- data.frame(y1 = ymat[, 1], y2 = ymat[, 2],
                    y3 = ymat[, 1], y4 = ymat[, 2],
                    x2 = runif(nn))
summary(bdata)
## Not run:  plot(ymat, col = "blue") 
fit1 <-  # 2 responses, e.g., (y1,y2) is the 1st
  vglm(cbind(y1, y2, y3, y4) ~ 1, fam = binormalcop,
       crit = "coef",  # Sometimes a good idea
       data = bdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
head(fitted(fit1))
summary(fit1)

# Another example; rho is a linear function of x2
bdata <- transform(bdata, rho = -0.5 + x2)
ymat <- rbinormcop(n = nn, rho = with(bdata, rho))
bdata <- transform(bdata, y5 = ymat[, 1], y6 = ymat[, 2])
fit2 <- vgam(cbind(y5, y6) ~ s(x2), data = bdata,
             binormalcop(lrho = "identitylink"), trace = TRUE)
## Not run: plot(fit2, lcol = "blue", scol = "orange", se = TRUE)