# bifrankcopUC: Frank's Bivariate Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Density, distribution function, and random generation for the (one parameter) bivariate Frank distribution.

## Usage

 ```1 2 3``` ```dbifrankcop(x1, x2, apar, log = FALSE) pbifrankcop(q1, q2, apar) rbifrankcop(n, apar) ```

## Arguments

 `x1, x2, q1, q2` vector of quantiles. `n` number of observations. Same as in `runif`. `apar` the positive association parameter. `log` Logical. If `log = TRUE` then the logarithm of the density is returned.

## Details

See `bifrankcop`, the VGAM family functions for estimating the association parameter by maximum likelihood estimation, for the formula of the cumulative distribution function and other details.

## Value

`dbifrankcop` gives the density, `pbifrankcop` gives the distribution function, and `rbifrankcop` generates random deviates (a two-column matrix).

T. W. Yee

## References

Genest, C. (1987). Frank's family of bivariate distributions. Biometrika, 74, 549–555.

`bifrankcop`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```## Not run: N <- 100; apar <- exp(2) xx <- seq(-0.30, 1.30, len = N) ox <- expand.grid(xx, xx) zedd <- dbifrankcop(ox[, 1], ox[, 2], apar = apar) contour(xx, xx, matrix(zedd, N, N)) zedd <- pbifrankcop(ox[, 1], ox[, 2], apar = apar) contour(xx, xx, matrix(zedd, N, N)) plot(rr <- rbifrankcop(n = 3000, apar = exp(4))) par(mfrow = c(1, 2)) hist(rr[, 1]); hist(rr[, 2]) # Should be uniform ## End(Not run) ```