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Also + $X_1 = 0 = X_2$ $\mathbb{P}$ f.s. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Setze $Y_i(x) \coloneqq \mathbbm{1}_{\{X_i \le x\}}$. Dann + ist $Y_i(x) \sim \text{Bernoulli}_{p}$ mit: + \[ + p := \mathbb{P}(X_i \le x) =\mathbb{P}(X_1 \le x) = \mathbb{F}(x) + .\] + Dann + ist $n \hat{\mathbb{F}}_n(x) = \sum_{i=1}^{n} Y_i(x)$ und damit + $n \hat{\mathbb{F}}_n(x) \sim \text{Bin}_{n,p}$ mit $p = \mathbb{P}(X_1 \le x) = \mathbb{F}(x)$. + Damit folgt + \begin{salign*} + \E(\hat{\mathbb{F}}_n(x)) &= \frac{1}{n} \E(n\hat{\mathbb{F}}_n(x)) + = \frac{1}{n} n p = p = \mathbb{F}(x)\\ + \var(\hat{\mathbb{F}}_n(x)) &= \frac{1}{n^2} \var(n \hat{\mathbb{F}}_n(x)) = \frac{p(1-p)}{n} + = \frac{\mathbb{F}(x)(1 - \mathbb{F}(x)}{n} + .\end{salign*} + \item Es ist $\hat{\mathbb{F}}_n(x) = \overline{Y}_n(x)$. Also folgt + \begin{salign*} + \lim_{n \to \infty} \hat{\mathbb{F}}_n(x) = + \lim_{n \to \infty} \overline{Y}_n(x) + &\stackrel{\text{SGGZ}}{=} \E(Y_1) = \mathbb{F}(x) + .\end{salign*} + \item Es gilt da $\var(Y_1) = p(1-p) = \mathbb{F}(x)(1 - \mathbb{F}(x))$: + \begin{salign*} + \sqrt{n} (\hat{\mathbb{F}}_n(x) - \mathbb{F}(x)) + = \sqrt{n} (\overline{Y}_n(x) - \mathbb{F}(x)) + \xrightarrow[\text{ZGWS}]{D} N_{(0, \var(Y_1(x))} + = N_{(0, \mathbb{F}(x)(1 - \mathbb{F}(x))} + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $b \in \R^{+}$. Dann ist + \begin{salign*} + \text{Bias}_b(\hat{b}_n) &= \E_b(\hat{b}_n - b) = 2 \E_b(\overline{X}_n) - b = + 2 \E_b(X_1) - b = 0 \\ + \var_b(\hat{b}_n) &= 4 \var_b(\overline{X}_n) = \frac{4}{n} \var_b(X_1) = \frac{b^2}{3n} + .\end{salign*} + Setze $Y_i \coloneqq 2 X_i$. Dann ist $\overline{Y}_n = 2 \overline{X}_n = \hat{b}_n$. Damit + folgt mit ZGWS direkt + \[ + \sqrt{n} (\hat{b}_n - b) = \sqrt{n} (\overline{Y}_n - \E(Y_1)) + \xrightarrow{D} N_{(0, \var(Y_1)} = N_{(0, b^2/3)} + .\] + \item Es ist $\hat{b}_n$ nach (a) $\sqrt{n} $-konsistent. Außerdem ist mit (a), dass + \[ + \frac{\sqrt{3n} }{b} (\hat{b}_n - b) \xrightarrow{D} N_{(0, 1)} + .\] Also ist $\frac{b}{\sqrt{3} }$ Störparameter. Da $\hat{b}_n = \overline{Y}_n$ + und $\E(Y_1) = b$, folgt mit dem SGGZ, dass + \[ + \hat{b}_n = \overline{Y}_n \xrightarrow{\mathbb{P} \text{ f.s.}} \E(Y_1) = b + .\] Also insbesondere $\hat{b}_n$ konsistenter Schätzer für $b$ und damit + auch $\frac{\hat{b}_n}{\sqrt{3} }$ konsistenter Schätzer für den Störparameter + $\frac{b}{\sqrt{3} }$. Damit folgt nach VL, dass + $C_n^{(b)} \coloneqq [\hat{b}_n - \frac{\hat{b}_n}{\sqrt{3n} }q_{1-\alpha}, \infty)$ + ein $1-\alpha$ KB für die richtigen Parameter $\mathcal{R}_{b} = [b, \infty)$ ist, wobei + $q_{1-\alpha}$ das $1-\alpha$ Quantil der Standardnormalverteilung ist. + \item $H_0$ ist die assoziierte Nullhypothese zu $\mathcal{R}_{b_0}$. Damit folgt nach + VL, da $C_n^{(b)}$ asympt. $1-\alpha$-KB, dass + \[ + \phi_n^{(b)} = \mathbbm{1}_{\{b_0 \not\in C_{n}^{(b)}\} } + = \mathbbm{1}_{\{ \hat{b}_n - \frac{\hat{b}_n}{\sqrt{3n} } q_{1-\alpha} > b_0\} } + \] ein asympt. $1-\alpha$ Test ist. + \item Ausrechnen ergibt $\overline{X}_n = 50.88$ und damit + \[ + \hat{b}_{10} - \frac{\hat{b}_{10}}{\sqrt{3 \cdot 10} } 1.64 = 71.28 < 100 + \qquad C_{10}^{(b)} = [71.28, \infty) + .\] Also sollte $H_0$ nicht abgelehnt werden. + \end{enumerate} +\end{aufgabe} + +\end{document}