# Level sets for bivariate normal, student-t and skew-normal distributions probability densities. Level sets of the bivariate normal, student-t and skew-normal distributions probability densities for a given probability. ```r ellipse(center=c(0,0), alpha=c(0,0), sigma=diag(2), df=1, prob=0.01, npoints=250, pos=FALSE) ``` ## Arguments - `center`: A vector of length 2 corresponding to the location of the distribution. - `alpha`: A vector of length 2 corresponding to the skewness of the skew-normal distribution. - `sigma`: A 2 by 2 variance-covariance matrix. - `df`: An integer corresponding to the degree of freedom of the student-t distribution. - `prob`: The probability level. See `details` - `npoints`: The maximum number of points at which it is evaluated. - `pos`: If `pos=TRUE` only the region on the positive quadrant is kept. ## Returns Returns a bivariate vector of $250$ rows if `pos=FALSE`, and half otherwise. ## Details The Level sets are defined as $R(f_\alpha)={ x: f(x) \geq f_\alpha }$ where $f_\alpha$ is the largest constant such that $P(X \in R(f_\alpha)) \geq 1-\alpha$. Here we consider $f(x)$ to be the bivariate normal, student-t or skew-normal density. ## Author(s) Simone Padoan, simone.padoan@unibocconi.it , [https://faculty.unibocconi.it/simonepadoan/](https://faculty.unibocconi.it/simonepadoan/); Boris Beranger, borisberanger@gmail.com [https://www.borisberanger.com](https://www.borisberanger.com); ## Examples ```r library(mvtnorm) # Data simulation (Bivariate-t on positive quadrant) rho <- 0.5 sigma <- matrix(c(1,rho,rho,1), ncol=2) df <- 2 set.seed(101) n <- 1500 data <- rmvt(5*n, sigma=sigma, df=df) data <- data[data[,1]>0 & data[,2]>0, ] data <- data[1:n, ] P <- c(1/750, 1/1500, 1/3000) ell1 <- ellipse(prob=1-P[1], sigma=sigma, df=df, pos=TRUE) ell2 <- ellipse(prob=1-P[2], sigma=sigma, df=df, pos=TRUE) ell3 <- ellipse(prob=1-P[3], sigma=sigma, df=df, pos=TRUE) plot(data, xlim=c(0,max(data[,1],ell1[,1],ell2[,1],ell3[,1])), ylim=c(0,max(data[,2],ell1[,2],ell2[,2],ell3[,2])), pch=19) points(ell1, type="l", lwd=2, lty=1) points(ell2, type="l", lwd=2, lty=1) points(ell3, type="l", lwd=2, lty=1) ```

ellipse function