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Propiedades de la distribución de Hotelling y de Wilks

Distribución de Hotelling

\[\Gamma^2(d,m)\stackrel{d}{=}\frac{md}{m-d+1}F_{d,m-d+1}.\]

Distribución de Wilks. Casos especiales

\[\Lambda(d,m_H,m_E)\stackrel{d}{=} \Lambda(m_H,d,m_E-d+m_H).\]

Si \(d=1\),

\[\frac{1-\Lambda(1,m_H,m_E)}{\Lambda(1,m_H,m_E)}\stackrel{d}{=} \frac{m_H}{m_E} F_{m_H,m_E}.\]

Si \(m_H=1\),

\[\frac{1-\Lambda(d,1,m_E)}{\Lambda(d,1,m_E)}\stackrel{d}{=} \frac{d}{m_E-d+1} F_{d,m_E-d+1}.\]

Si \(d=2\) (\(m_H\geq 2\)),

\[\frac{1-\sqrt{\Lambda(2,m_H,m_E)}}{\sqrt{\Lambda(2,m_H,m_E)}}\stackrel{d}{=} \frac{m_H}{m_E-1} F_{2m_H,2(m_E-1)}.\]

Si \(m_H=2\) (\(d\geq 2\)),

\[\frac{1-\sqrt{\Lambda(d,2,m_E)}}{\sqrt{\Lambda(d,2,m_E)}}\stackrel{d}{=} \frac{d}{m_E-d+1} F_{2d,2(m_E-d+1)}.\]

Aproximación de Bartlett: Para \(m_E\) grande,

\[-f\ln \Lambda(d,m_H,m_E)\ \mbox{se aproxima a}\ \chi_{dm_H}^2\] donde

\(f=m_E-\frac{1}{2}(d-m_H+1).\)

Aproximación de Rao:

\[\frac{1-\Lambda(d,m_H,m_E)^{1/t}}{\Lambda(d,m_H,m_E)^{1/t}}\frac{ft-g}{dm_H}\textnormal{ es aproximadamente } F_{dm_H,ft-g}\] siendo

\[f=m_E-\frac{1}{2}(d-m_H+1),\]

\[t=\left(\frac{d^2m_H^2-4}{d^2+m_H^2-5}\right)^{1/2},\]

\[g=\frac{dm_H-2}{2}.\]