Statistik 3 - Beispiel 5

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Zu zeigen ist TSS = ESS + RSS

Failed to parse (unknown error): \mathrm{TSS} = \sum (y_i - \bar y)^2


Failed to parse (unknown error): \mathrm{ESS} = \sum (\hat{y}_i - \bar y)^2


Failed to parse (unknown error): \mathrm{RSS} = \sum (y_i - \hat{y}_i)^2 = \sum \hat{u}_i^2


Failed to parse (unknown error): \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:

Failed to parse (unknown error):  y_i = \hat{y}_i + \hat{u}_i 


Failed to parse (unknown error):  \hat{u}_i = y_i - \hat{y}_i 


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


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


Failed to parse (unknown error): \sum (y_i - \bar y)^2 =


Failed to parse (unknown error): \sum( \hat{y}_i + \hat{u}_i - \bar{y}_i )^2 =


Failed to parse (unknown error): \sum( (\hat{y}_i - \bar{y}_i) + \hat{u}_i )^2 =


Failed to parse (unknown error): \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}


Failed to parse (unknown error): \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}

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