Statistik 3 - Prüfung - Formeln

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Lineare Regressionsmodell

Schätzer a, b

\hat a = \bar y - \hat b \bar x \,

\hat b = \frac{s_{x,y}}{s_{x,x}} \,

Identitäten

\hat{y}_i = \hat a + \hat b x_i = \bar y - \hat b \bar x + \hat b x_i = \bar y + \hat b (x_i - \bar x)

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

Eigenschaften


\sum_{t=1}^n \hat{u}_t^2 = 
\sum_{t=1}^n (y_i - \hat{y}_i)^2 = 
\sum_{t=1}^n (y_i - \bar y - \hat b (x_i - \bar x))^2 = 
\sum_{t=1}^n (y_i - \bar y)^2 - 2 \hat b \sum_{t=1}^n (y_i - \bar y)(x_i - \bar x) + \hat{b}^2 \sum_{t=1}^n (x_i - \bar x)^2 = 



= (n-1) \left( s_{y,y} - 2 \frac{s_{x,y}}{s_{x,x}} s_{x,y} + \frac{s_{x,y}^2}{s_{x,x}^2} s_{x,x} \right) =
(n-1) \left( s_{y,y} - \frac{s_{x,y}^2}{s_{x,x}} \right) =
(n-1) ( s_{y,y} - \hat b^2 s_{x,x} ) =
(n-1) ( s_{y,y} - \hat b s_{x,y} )

\hat{\sigma}^2 = \frac{1}{n - 2} \sum_{t=1}^n \hat{u}_t^2 = \frac{n-1}{n - 2} ( s_{y,y} - \hat b s_{x,y} )


\hat{\sigma}_{\hat{\beta}}^2 = \frac{\hat{\sigma}^2}{(n-1) s_{x,x}} = \frac{1}{n-2} \left( \frac{s_{y,y}}{s_{x,x}} - \hat{b}^2 \right) 
X'\hat u = 0 \, Statistik 3 - Beispiel 27#Beispiel 27 a 1
\hat{y}^' \hat{u} = 0 \, Statistik 3 - Beispiel 27#Beispiel 27 a 2
\hat u = y - \hat{y} = y - X \hat\beta = y - X(X'X)^{-1}X'y = (I_n - X(X'X)^{-1}X') y \,
M = I_n - X(X'X)^{-1}X' \,

M = M'\, (symmetrisch)

M^2 = M\, (idempotent)

MX = 0 \,
M \hat u = M (X\beta + u) = M X \beta + M u = M u \,

Tests

T = \frac{\hat \beta - \hat{\beta}^{\star}}{\hat{\sigma}_{\hat{\beta}}} mit n - 2\, Freiheitsgraden

Sonstiges

\operatorname{VC}(A \Sigma) = A \operatorname{VC}(\Sigma) A'

Normalverteilung

Reproduktionssatz: Z \sim N(\mu, \Sigma), W = A Z + a \Rightarrow W \sim N(A \mu + a, A \Sigma A')  falls A\, vollen zeilen Rang hat.

Matrix Rechen Regeln

Trace bzw. Spur

\operatorname{tr}(A) = \sum_{i=1}^n a_{i,i} \,

\operatorname{tr}(A + B) = \operatorname{tr}(A) + \operatorname{tr}(B) \,

\operatorname{tr}(A') = \operatorname{tr}(A)\,

\operatorname{tr}(A B) = \operatorname{tr}(B A) \, für A_{n \times k} \, und B_{k \times n}\,

Beweis siehe Statistik 3 - Beispiel 20