Esquema del mètode
modifica
En els models de regressió lineal estàndard s'observen dades
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{\displaystyle \{y_{i},x_{ij}\}_{i=1,\dots ,n,j=2,\dots ,k}}
n unitats estadístiques .[3]
Els valors de resposta es col·loquen en un vector,
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{\displaystyle \mathbf {y} \equiv {\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}},}
matriu de disseny ,
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{\displaystyle \mathbf {X} \equiv {\begin{pmatrix}1&x_{12}&x_{13}&\cdots &x_{1k}\\1&x_{22}&x_{23}&\cdots &x_{2k}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n2}&x_{n3}&\cdots &x_{nk}\end{pmatrix}},}
k
{\displaystyle k}
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{\displaystyle i}
mitjana condicional de
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{\displaystyle \mathbf {y} }
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{\displaystyle \mathbf {X} }
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{\displaystyle \mathbf {X} }
variància condicional del terme d'error donat
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{\displaystyle \mathbf {X} }
matriu de covariància no singular coneguda,
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{\displaystyle \mathbf {\Omega } }
[4]
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{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\quad \operatorname {E} [{\boldsymbol {\varepsilon }}\mid \mathbf {X} ]=0,\quad \operatorname {Cov} [{\boldsymbol {\varepsilon }}\mid \mathbf {X} ]={\boldsymbol {\Omega }},}
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{\displaystyle {\boldsymbol {\beta }}\in \mathbb {R} ^{k}}
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{\displaystyle \mathbf {b} }
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{\displaystyle {\boldsymbol {\beta }}}
residual per
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{\displaystyle \mathbf {b} }
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{\displaystyle \mathbf {y} -\mathbf {X} \mathbf {b} }
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{\displaystyle {\boldsymbol {\beta }}}
longitud al quadrat de Mahalanobis d'aquest vector residual:
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{\displaystyle {\begin{aligned}{\hat {\boldsymbol {\beta }}}&={\underset {\mathbf {b} }{\operatorname {argmin} }}\,(\mathbf {y} -\mathbf {X} \mathbf {b} )^{\mathrm {T} }\mathbf {\Omega } ^{-1}(\mathbf {y} -\mathbf {X} \mathbf {b} )\\&={\underset {\mathbf {b} }{\operatorname {argmin} }}\,\mathbf {y} ^{\mathrm {T} }\,\mathbf {\Omega } ^{-1}\mathbf {y} +(\mathbf {X} \mathbf {b} )^{\mathrm {T} }\mathbf {\Omega } ^{-1}\mathbf {X} \mathbf {b} -\mathbf {y} ^{\mathrm {T} }\mathbf {\Omega } ^{-1}\mathbf {X} \mathbf {b} -(\mathbf {X} \mathbf {b} )^{\mathrm {T} }\mathbf {\Omega } ^{-1}\mathbf {y} \,,\end{aligned}}}
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{\displaystyle {\hat {\boldsymbol {\beta }}}={\underset {\mathbf {b} }{\operatorname {argmin} }}\,\mathbf {y} ^{\mathrm {T} }\,\mathbf {\Omega } ^{-1}\mathbf {y} +\mathbf {b} ^{\mathrm {T} }\mathbf {X} ^{\mathrm {T} }\mathbf {\Omega } ^{-1}\mathbf {X} \mathbf {b} -2\mathbf {b} ^{\mathrm {T} }\mathbf {X} ^{\mathrm {T} }\mathbf {\Omega } ^{-1}\mathbf {y} ,}
![{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\quad \operatorname {E} [{\boldsymbol {\varepsilon }}\mid \mathbf {X} ]=0,\quad \operatorname {Cov} [{\boldsymbol {\varepsilon }}\mid \mathbf {X} ]={\boldsymbol {\Omega }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/683642aa3b58460b7a5b540950bac4037f946511)
