Empirische Momente von zufälligen und deterministischen Samples

Erzeugung von Samples

$$ y_i = m + \sigma \cdot w_i \quad w_i \in \mathcal{N}(0, 1) \quad i = 1, \dots , L $$

• $w_i$: Grundsamples
• $m$: Mittelwert

Check:

$$ \begin{aligned} &E\left\{y_{i}\right\}=E\left\{m+\sigma \cdot w_{i}\right\}=E\{m\}+\sigma \cdot E\left\{w_{i}\right\}=m \\ &E\left\{\left(y_{i}-m\right)^{2}\right\}=E\left\{\left(\sigma \cdot w_{i}\right)^{2}\right\}=\sigma^{2} E\left\{w_{i}^{2}\right\}=\sigma^{2} \end{aligned} $$

Empirische Schätzer

$$ \hat{m}=\frac{1}{L} \sum_{i=1}^{L} y_{i} $$ $$ \begin{aligned} \hat{c}=\hat{\sigma}^{2} &=\frac{1}{L} \sum_{i=1}^{L}\left(y_{i}-\hat{m}\right)^{2} \\ &=\frac{1}{L} \sum_{i=1}^{L}\left(y_{i}^{2}-2 \hat{m} y_{i}+\hat{m}^{2}\right) \\ &=\frac{1}{L} \sum_{i=1}^{L} y_{i}^{2}-\left(\frac{1}{L} \sum_{i=1}^{L} y_{i}\right)^{2} \\ &=\frac{1}{L} \sum_{i=1}^{L} y_{i}^{2}-\frac{1}{L^{2}} \sum_{i=1}^{L} \sum_{j=1}^{L} y_{i} y_{j} \end{aligned} $$

Überprüfung

• Mittelwert

  $$ \begin{aligned} E\{\hat{m}\} &=E\left\{\frac{1}{L} \sum_{i=1}^{L}\left(m+2 w_{i}\right)\right\} \\ &=\frac{1}{L} \sum_{i=1}^{L} E\left\{m_{i}+w_{i}\right\} \\ &=m \quad ✅ \end{aligned} $$

• Varianz

  $$ \begin{aligned} E\{\hat{C}\}&=E\left\{\frac{1}{L} \sum_{i=1}^{L}\left(m+\sigma w_{i}\right)^{2}-\frac{1}{L^{2}} \sum_{i=1}^{l} \sum_{j=1}^{c}\left(m_{1}+\sigma w_{i}\right)\left(m+\sigma w_{i}\right)\right\}\\ &=\frac{1}{L} \sum_{i=1}^{L} E\left\{m^{2}+2\sigma w_{i}+\sigma^{2} w_{i}^{2}\right\}- \\ & \qquad\frac{1}{L^{2}} \sum_{i=1}^{L} \sum_{j=1}^{L} E\left\{m^{2}+m \sigma w_{i}+m \sigma w_{j}+\sigma^{2} w_{i} w_{j}\right\}\\ &=\frac{1}{L} \sum_{i=1}^{L}\left(m^{2}+\sigma^{2}\right)-\frac{1}{L^{2}} \sum_{i=1}^{L} \sum_{j=1}^{L}\left(m^{2}+\sigma^{2} E\left\{\omega_{i} \omega_{j}\right\}\right)\\ &=m^{2}+\sigma^{2}-m^{2}-\frac{1}{L^{2}} \cdot L \cdot{\sigma^{2}}^{2}\\ &=\sigma^{2}-\frac{1}{L} \sigma^{2}\\ &=\frac{L-1}{L} \cdot \sigma^{2} \end{aligned} $$

Für deterministische Samples (z.B.)

$$ \begin{aligned} &y_{1}=m-\sigma \\ &y_{2}=m+\sigma \end{aligned} $$ $$ \begin{aligned} \hat{m} &=\frac{1}{2}(m-\sigma+m+\sigma)=m \quad ✅ \\ \hat{z}^{2} &=\frac{1}{2}\left[(m-\sigma)^{2}+(m+\sigma)^{2}\right]-\frac{1}{4}(m-\sigma+m+\sigma)^{2} \\ &=\frac{1}{2}\left[m^{2}-2 m \sigma+\sigma^{2}+m^{2}+2 m \sigma+\sigma^{2}\right]-m^{2} \\ &=\frac{1}{2}\left[2 m^{2}+2\sigma^{2}\right]-m^{2} \\ &=\sigma^{2} \end{aligned} $$