Statistik 3 - Beispiel 13

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Bsp. 13

Schätzer \check b  = \frac{\bar y}{\bar x} \tilde b  = \frac{\overline{x y}}{\overline{x x}} \hat b  = \frac{\overline{x y} - \bar x \bar y}{\overline{x x} - \bar x \bar x}
linear (a) \check b = \sum c_i y_i = \frac{\sum y_i}{\sum x_i} \Rightarrow c_i = \frac{1}{\sum x_i} \tilde b = \frac{\sum x_i y_i}{\sum x_i^2} \Rightarrow c_i = \frac{x_i}{\sum x_i^2} \hat b = \frac{\sum (x_i - \bar x) y_i }{\sum (x_i - \bar x)^2} \Rightarrow c_i = \frac{x_i - \bar x}{\sum (x_i - \bar x)^2}
unverzerrt (b) \operatorname{E}(\check b) = \operatorname{E}\left(\frac{ \sum b x_i + u_i } { \sum x_i } \right) = \frac{b \sum x_i + \operatorname{E}(u_i)}{\sum x_i} = b \operatorname{E}(\tilde b) = b Statistik 3 - Beispiel 11 \operatorname{E}(\hat b) = \operatorname{E}\left(\frac{\sum (x_i - \bar x) y_i }{ \sum (x_i - \bar x)^2 } \right) =

= \operatorname{E}\left(\frac{\sum (x_i - \bar x) (b x_i + u_i }{ \sum (x_i - \bar x)^2 } \right) =

= \operatorname{E}\left(\frac{b \sum x_i (x_i - \bar x) + \sum x_i u_i + \bar x \sum u_i }{ \sum (x_i - \bar x)^2 } \right) =

= \frac{b \sum x_i (x_i - \bar x) + \sum x_i \operatorname{E}(u_i) + \sum \bar x \operatorname{E}(u_i) }{ \sum (x_i - \bar x)^2 } =

= \frac{b \sum x_i (x_i - \bar x) }{ \sum (x_i - \bar x)^2 } =

= b \frac{\sum (x_i - \bar x)^2 }{ \sum (x_i - \bar x)^2 } = b Kovarianz von x\,

Varianz (d) \operatorname{Var}(\check b) = 
\operatorname{E}\left( \left(\frac{\bar y}{\bar x} - b \right)^2 \right) =  
\operatorname{E}\left( \left(\frac{b \bar x + \bar u }{\bar x} - b \right)^2 \right) =


=\operatorname{E}\left( \left(\frac{b \bar x + \bar u - b \bar x}{\bar x} \right)^2 \right) = 
\frac{\operatorname{E}(\bar u^2)}{\bar x} = 
\frac{\sigma^2}{\sum x_i}

\operatorname{Var}(\tilde b) = \frac{\sigma^2}{\sum x_i^2} \operatorname{Var}(\hat b) = \operatorname{E}\left( \left( \frac{\sum (x_i - \bar x) y_i }{ \sum (x_i - \bar x)^2 } - b \right)^2 \right) =

 = \operatorname{E}\left( \left( \frac{\sum (x_i - \bar x) (b x_i + u_i }{ \sum (x_i - \bar x)^2 } - b \right)^2 \right) =

 = \operatorname{E}\left( \left( \frac{ b \sum x_i (x_i - \bar x) + \sum u_i (x_i - \bar x) - b \sum (x_i - \bar x)^2 }{ \sum (x_i - \bar x)^2 } \right)^2 \right) =

 = \operatorname{E}\left( \left( \frac{ b \sum (x_i - \bar x)^2 + \sum u_i (x_i - \bar x) - b \sum (x_i - \bar x)^2 }{ \sum (x_i - \bar x)^2 } \right)^2 \right) =

 = \operatorname{E}\left( \left( \frac{ \sum u_i (x_i - \bar x) }{ \sum (x_i - \bar x)^2 } \right)^2 \right) =

 = \operatorname{E}\left( \frac{ \sum u_i^2 (x_i - \bar x)^2 + \sum_{i \ne j} u_i u_j (x_i - \bar x) (x_j - \bar x)}{ \sum (x_i - \bar x)^2 } \right) =

 = \frac{ \sum \operatorname{E}(u_i^2) (x_i - \bar x)^2 + \sum_{i \ne j} \operatorname{E}(u_i u_j) (x_i - \bar x) (x_j - \bar x)}{ \sum (x_i - \bar x)^2 } = Wichtig: \operatorname{E}(u_i u_j) = 0, i \ne j

 = \sigma^2 \frac{ \sum (x_i - \bar x)^2 }{ \sum (x_i - \bar x)^2 } = \sigma^2

Bsp. 13 (e)

\tilde b\, ist der beste Schätzer, laut Gauss-Markov für das Modell y_i = b x_i + u_i \,.

Bsp. 13 (f)

\hat b\, ist der beste Schätzer, laut Gauss-Markov für das Modell y_i = a + b x_i + u_i \,.