Statistik 3 - Beispiel 5

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Zu zeigen ist TSS = ESS + RSS

\mathrm{TSS} = \sum (y_i - \bar y)^2

\mathrm{ESS} = \sum (\hat{y}_i - \bar y)^2

\mathrm{RSS} = \sum (y_i - \hat{y}_i)^2 = \sum \hat{u}_i^2

 \sum (y_i - \bar y)^2 = \sum (\hat{y}_i - \bar y)^2 + \sum (y_i - \hat{y}_i)^2

Um TSS umzuformen sollten wir uns noch anfolgendes erinnern:

 y_i = \hat{y}_i + \hat{u}_i 

 \hat{u}_i = y_i - \hat{y}_i 

 \sum \hat{u}_i = 0  wegen Normalgleichung:  \sum (y_i - \hat{y}_i) =  \sum (y_i - a - b x_i) = 0 

 \sum \hat{u}_i x_i = 0  wegen Normalgleichung:  \sum (y_i - \hat{y}_i) x_i = \sum (y_i - a - b x_i) x_i = 0 


 \sum (y_i - \bar y)^2  =

 \sum( \hat{y}_i + \hat{u}_i - \bar{y}_i )^2 =

 \sum( (\hat{y}_i - \bar{y}_i) + \hat{u}_i )^2 =

 
 \begin{matrix}
   \mathrm{TSS} = \underbrace{\sum(\hat{y}_i - \bar{y}_i)^2} & + & \underbrace{\sum(\hat{u}_i )^2} & + 2 & \underbrace{\sum(\hat{u}_i ( \hat{y}_i - \bar y ))} &  = 
   \sum(\hat{y}_i - \bar{y}_i)^2 + \sum(\hat{u}_i )^2 \\
   \mathrm{ESS} & & \mathrm{RSS} & & \sum \hat{u}_i \hat{y}_i - \bar y \sum \hat{u}_i & 
 \end{matrix}


 \begin{matrix}
   \sum \hat{u}_i \hat{y}_i - \bar y & \underbrace{\sum \hat{u}_i} &  = 
   \sum \hat{u}_i (a + b x_i) = a & \underbrace{\sum \hat{u}_i} & + b & \underbrace{\sum \hat{u_i} x_i} & = 0 \\
   & 0 & & 0 & & 0 &
 \end{matrix}