copula (version 1.1-7)

prob: Computing Probabilities of Hypercubes

Description

Compute probabilities of a \(d-\)dimensional random vector \(U\) distributed according to a given copula x to fall in a hypercube \((l,u]\), where \(l\) and \(u\) denote the lower and upper corners of the hypercube, respectively.

volume(<function>, *) computes the \(d-\)dimensional integral over the hypercube for “arbitrary” functions.

Usage

prob(x, l, u)
volume(object, ...) # generic function
# S3 method for function
volume(object, lower, upper, ...)

Value

prob() returns a numeric in \([0,1]\)

which is the probability \(P(l_i< U_i \le u_i)\).

volume() is the workhorse underlying prob() and its output is the \(d\)-volume, i.e., \(d\)-dimensional integral over the given hypercube, of the provided function.

Arguments

x

copula of dimension \(d\), that is, an object inheriting from Copula.

l, u, lower, upper

\(d\)-dimensional, numeric, lower and upper hypercube boundaries, respectively, satisfying \(0 \le l_i \le u_i \le 1\), for \(i\in{1,\dots,d}\). No checks are done for volume() except for being of the same length.

object

a function to compute the volume of wrt to the specified hypercube.

...

additional arguments passed to the function object.

See Also

pCopula(.).

Examples

## Construct a three-dimensional nested Joe copula with parameters
## chosen such that the Kendall's tau of the respective bivariate margins
## are 0.2 and 0.5.
theta0 <- copJoe@iTau(.2)
theta1 <- copJoe@iTau(.5)
C3 <- onacopula("J", C(theta0, 1, C(theta1, c(2,3))))

## Compute the probability of a random vector distributed according to
## this copula to fall inside the cube with lower point l and upper
## point u.
l <- c(.7,.8,.6)
u <- c(1,1,1)
prob(C3, l, u)

## ditto for a bivariate normal copula with rho = 0.8 :
Cn <- normalCopula(0.8)
(prob(Cn, c(.2,.4), c(.3,.6)) -> pr) # 0.0222...
## prob() just using volume(), internally:
pr == volume(function(z) pCopula(z, Cn), c(.2,.4), c(.3,.6))