User:Bruguiea sandbox

See real user User:Bruguiea. This page is for me to experiment without cheating on my edit count.

$$ (n; 1) $$ $$ (n; r) $$ $$ (r; 1) $$

$$ \hat{\beta} = (X' X)^{-1} X' y $$

$$ \hat{\epsilon} = \left(I - X (X' X)^{-1} X' \right) y $$

$$ t_{n-r} = \frac{\hat{\beta _i}}{ \sqrt{y' \left(I - X (X' X)^{-1} X' \right)' \left(I - X (X' X)^{-1} X' \right) y}} $$

$$ C = ER + \alpha. Risk $$

$$ \nu = \gamma \Beta _0 $$

$$ \delta $$

$$ x = \delta \star h $$

$$ \alpha _1 x_1 + \cdots + \alpha _n x_n \leftrightarrow \beta _1 y_1 + \cdots + \beta _p y_p $$

$$ J = \| y - X \beta \|^2 + h^2 \| \beta \| ^2 $$

$$ \frac{\partial J}{\partial \beta} = 0 \Leftrightarrow \hat{\beta} = \left( X' X + h^2 I\right)^{-1} X' y $$

$$ G = \frac{\operatorname{RSS}}{\tau ^2} = \frac{\| y - X \hat{\beta} \|^2}{ \left( \operatorname{Tr} \left( I - X (X' X + h^2 I) ^{-1} X' \right) \right) ^2} $$

$$ \beta = \Sigma _{22} ^ {-1} \Sigma _{2 U_k} $$

$$ \beta = \Sigma _{22} ^ {-1} \Sigma _{21} a^{(k)} $$

$$ \beta = \Sigma _{22} ^ {-1} \Sigma _{21} \Sigma _{11} ^{-1/2} c^{(k)} $$

$$ \beta = \Sigma _{22} ^ {-1/2} \Sigma _{22} ^ {-1/2} \Sigma _{21} \Sigma _{11} ^{-1/2} c^{(k)} $$

$$ R_{y_1 y_2} (t) = h * h ^{(-)} * R_{x_1 x_2} (t) $$