Pn_logtDensity-class {multIntTestFunc}

An S4 class to represent the function (\prod_{i=1}^n x_i^{-1})\frac{\Gamma\left[(\nu+n)/2\right]}{\Gamma(\nu/2)\nu^{n/2}\pi^{n/2}\left|{\Sigma}\right|^{1/2}}\left[1+\frac{1}{\nu}({\log(\vec{x})}-{\vec{\delta}})^{T}{\Sigma}^{-1}({\log(\vec{x})}-{\vec{\delta}})\right]^{-(\nu+n)/2} on [0,\infty)^n

Description

Implementation of the function

f \colon [0,\infty)^n \to (0,\infty),\, \vec{x} \mapsto f(\vec{x}) = (\prod_{i=1}^n x_i^{-1})\frac{\Gamma\left[(\nu+n)/2\right]}{\Gamma(\nu/2)\nu^{n/2}\pi^{n/2}\left|{\Sigma}\right|^{1/2}}\left[1+\frac{1}{\nu}({\log(\vec{x})}-{\vec{\delta}})^{T}{\Sigma}^{-1}({\log(\vec{x})}-{\vec{\delta}})\right]^{-(\nu+n)/2},

where n \in \{1,2,3,\ldots\} is the dimension of the integration domain [0,\infty)^n = \times_{i=1}^n [0,\infty). In this case the integral is know to be

\int_{[0,\infty)^n} f(\vec{x}) d\vec{x} = 1.

Details

The instance needs to be created with four parameters representing the dimension n, the location vector \vec{\delta}, the variance-covariance matrix \Sigma which needs to be symmetric positive definite and the degrees of freedom parameter \nu.

Slots

dim

An integer that captures the dimension

delta

A vector of size dim with real entries.

sigma

A matrix of size dim x dim that is symmetric positive definite.

df

A positive numerical value representing the degrees of freedom.

Author(s)

Klaus Herrmann

Examples

n <- as.integer(3)
f <- new("Pn_logtDensity",dim=n,delta=rep(0,n),sigma=diag(n),df=3)