Gaussian Copula (Bivariate) Family Function
Estimate the correlation parameter of the (bivariate) Gaussian copula distribution by maximum likelihood estimation.
binormalcop(lrho = "rhobitlink", irho = NULL, imethod = 1,
parallel = FALSE, zero = NULL)lrho, irho, imethod |
Details at |
parallel, zero |
Details at |
The cumulative distribution function is
P(Y1 <= y1, Y2 <= y2) = Phi_2(Φ^(-1)(y_1), Φ^(-1)(y_2); ρ)
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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
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.
T. W. Yee
Schepsmeier, U. and Stober, J. (2014). Derivatives and Fisher information of bivariate copulas. Statistical Papers 55, 525–542.
nn <- 1000
ymat <- rbinormcop(n = 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 <- vglm(cbind(y1, y2, y3, y4) ~ 1, # 2 responses, e.g., (y1,y2) is the first
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, las = 1)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.