Overview and literature

See also: Hahn and Hendrickson [377], Narula [445], Stoline and Ury [533], Tong [855], Genz et al. [362].

See also: Bechhofer and Dunnett [31]

See also: Genz and Bretz [361]

See also: Genz et al. [362]

See also: Hahn and Hendrickson [377]

See also: Narula [445]

See also: Stoline and Ury [533]

See also: Tong [855]

For tables see Soong [531].

For tables see David [204].

For tables see David et al. [205].

An \(n\)-dimensional random variable \(\textbf{X}\) with mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\) is said to have a nonsingular multivariate normal distribution, in symbols \(\boldsymbol{X} \sim \mathcal{N}_n(\boldsymbol{\mu}, \boldsymbol{\Sigma})\), if \(\boldsymbol{\Sigma}\) is positive definite, and the density function of \(\textbf{X}\) is of the form (see Tong 1990):

\[f(\boldsymbol{x; \mu, \Sigma}) = \frac{1}{(2\pi)^{n/2} \vert \boldsymbol{\Sigma} \vert ^{1/2}} e^{-Q_n(\boldsymbol{x; \mu, \Sigma})/2}, \quad \boldsymbol{x} \in \Re^n\]

where

\[Q_n(\boldsymbol{x; \mu, \Sigma}) = (\boldsymbol{x - \mu})' \boldsymbol{\Sigma^{-1}} (\boldsymbol{x - \mu}).\]

The notion of cumulative distribution function (cdf) in one dimension can be extended to the multidimensional case, based on rectangular regions. We define the cdf \(F(\mathbf {x} )\) of a random vector \(\mathbf {X}\) as the probability that all components \(\mathbf {X}\) are less than or equal to the corresponding values in the vector \(\mathbf {x}\):

\[F(\mathbf {x} )=\mathbb {P} (\mathbf {X} \leq \mathbf {x} ),\quad {\text{where }}\mathbf {X} \sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }}).\]

See also: Genz(2009), Bretz(2003).

Let \(\boldsymbol{R} = (\rho_{ij})\) be an \(n \times n\) symmetric matrix such that it is either positive definite or positive semidefinite and \(\rho_{ii} = 1 (i=1,\ldots,n)\). Let \(\textbf{Z} = (Z_1,\ldots,Z_{n})'\) have an \(\mathcal{N}_n(\boldsymbol{0}, \boldsymbol{R})\) distribution, and let the univariate variable \(S\) be such that \(S\) is independent of \(\boldsymbol{Z}\), and \(\nu S^2\) has a \(\chi^2(\nu)\) distribution. Then a natural generalization of the Student’s \(t\) variable is

\[\boldsymbol{t} = (t_1,\ldots,t_n)' = \left(\frac{Z_1}{S},\ldots,\frac{Z_n}{S}\right)' .\]

If \(\boldsymbol{R}\) is positive definite, then the density of \(\boldsymbol{t}\) (with correlation matrix \(\boldsymbol{R}\) and degrees of freedom \(\nu\)) is given by citep{Tong_1990}:

\[h(\boldsymbol{t; R}, \nu)= \frac{\Gamma((n+\nu)/2)}{(\nu \pi)^{n/2} \Gamma(\nu/2) \vert \boldsymbol{R} \vert^{1/2}} \left(1+\frac{1}{\nu} \boldsymbol{t}' \boldsymbol{R}^{-1} \boldsymbol{t} \right)^{-(n+\nu)/2} , \quad \boldsymbol{t} \in \Re^n.\]

The notion of cumulative distribution function (cdf) in one dimension can be extended to the multidimensional case, based on rectangular regions. We define the cdf \(F(\mathbf {x} )\) of a random vector \(\mathbf {X}\) as the probability that all components \(\mathbf {X}\) are less than or equal to the corresponding values in the vector \(\mathbf {x}\):

\[F(\mathbf {x} )=\mathbb {P} (\mathbf {X} \leq \mathbf {x} ),\quad {\text{where }}\mathbf {X} \sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }}).\]

See also: Genz(2009), Bretz(2003).