# Gaussian Copula (Bivariate) Family Function Estimate the correlation parameter of the (bivariate) Gaussian copula distribution by maximum likelihood estimation. ```r 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 $$ P(Y_1 \leq y_1, Y_2 \leq y_2) = \Phi_2( \Phi^{-1}(y_1), \Phi^{-1}(y_2); \rho )%P(Y1 <= y1, Y2 <= y2) =Phi_2(\Phi^(-1)(y_1), \Phi^(-1)(y_2); \rho) $$ 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. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## References Schepsmeier, U. and Stober, J. (2014). Derivatives and Fisher information of bivariate copulas. **Statistical Papers** 55 , 525--542. ## Author(s) T. W. Yee ## 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. ## See Also `rbinormcop`, `rhobitlink`, `pnorm`, `kendall.tau`. ## Examples ```r 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) ```

binormalcop function