ellipse {ExtremalDep}

Level sets for bivariate normal, student-t and skew-normal distributions probability densities.

Description

Level sets of the bivariate normal, student-t and skew-normal distributions probability densities for a given probability.

Usage

	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.

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.

Value

Returns a bivariate vector of 250 rows if pos=FALSE, and half otherwise.

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail] https://www.borisberanger.com;

Examples

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)