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z2t#+f%qa5pcW4V;m-AWa2V71wO7{u?rTK&3S2k^3rvGh}9m&-zA7U+YK(5F_mcy@p6i^zF$8+OA`HdPt5q@yc3!lBlcVDw=|Np_$^fS zPDS90e$ED56PXaHS0qD@J4^x_gd4>L*GL=NZ+Sqt9B3xiY7t(s1A$ImXcw>86TZ}T z2iJW)i26Av6l~3(Q;9g!7!=Ag(FBwAB&337u~(erG4b?4upM+Kn3E6Zyd*r&b07fE z|1JzHh(O?l@Th diff --git a/wtheo5.tex b/wtheo5.tex index 0aff6ea..2f7f1d3 100644 --- a/wtheo5.tex +++ b/wtheo5.tex @@ -1,4 +1,4 @@ -\documentclass[uebung]{../../../lecture} +\documentclass[uebung]{lecture} \title{Wtheo 0: Übungsblatt 5} \author{Josua Kugler, Christian Merten} @@ -41,8 +41,49 @@ \end{proof} \end{aufgabe} -\stepcounter{aufgabe} - +\begin{aufgabe}[] + \begin{enumerate}[(a)] + \item Der Neyman-Pearson-Test $\varphi \colon \R \to \{1,0\}$ ist gegeben durch $\mathbbm{1}_{A_{k_\alpha}}$ mit + \[ + A_{k_\alpha} = \{x \in \R | \mathbbm{f}_1(x) \geq k_\alpha \mathbbm{f}_0(x)\} = \{x \in \R | \mathbbm{f}_1(x) / \mathbbm{f}_0(x) \geq k_\alpha\}, + \] wobei wir zeigen werden, dass ein $k_\alpha$ existiert mit $\mathbbm{P}_0(A_{k_\alpha}) = \alpha$. + Dabei ist der Likelihoodquotient $L(x) = \frac{\mathbbm{f}_1(x)}{\mathbbm{f}_0(x)} = \frac{\lambda_0}{\lambda_1} e^{\left(\frac{1}{\lambda_0} - \frac{1}{\lambda_1}\right) x}$ stetig und wegen $\lambda_0 < \lambda_1 \implies \frac{1}{\lambda_0} > \frac{1}{\lambda_1}$ auch monoton wachsend. Insbesondere existiert zu jedem $k_\alpha$ genau ein $c_\alpha$ mit + \[ + A_{k_\alpha} = [c_\alpha, \infty). + \] + Dann gilt + \[ + \alpha \overset{!}{=} \mathbbm{P}_0(A_{k_\alpha}) = \int_{A_{k_\alpha}} \mathbbm{f}_0(x) \d{x} = \int_{c_\alpha}^\infty \frac{1}{\lambda_0}e^{- \frac{1}{\lambda_0} x} \d{x} = e^{- \frac{1}{\lambda_0} c_\alpha} + \] + Daher gilt $c_\alpha = -\lambda_0 \log(\alpha)$ und damit $\varphi = \mathbbm{1}_{[-\lambda_0\log(\alpha), \infty)}$ + \item Offensichtlich ist also $\varphi$ unabhängig von $\lambda_1$. + Sei $\lambda < \lambda_0$. Dann gilt + \[ + \mathbbm{P}_\lambda(A_{k_\alpha}) = \int_{-\lambda_0\log(\alpha)}^\infty \frac{1}{\lambda}e^{-\frac{1}{\lambda}x} \d{x} = e^{\frac{\lambda_0}{\lambda}\log(\alpha)} = \alpha^{\frac{\lambda_0}{\lambda}} < \alpha, + \] + da $\alpha < 1$. + Damit hält $\varphi$ für $H_0: \lambda \le \lambda_0$ das Signifikanzniveau $\alpha$ ein und ist folglich gleichmäßig bester Test mit $H_1: \lambda > \lambda_0$, da für alle $\lambda_1 > \lambda_0$ nach dem Neyman-Pearson-Lemma $\varphi$ ein bester Test ist. + \item Die Forscher liegen falsch, wenn die Nullhypothese $\lambda \leq \lambda_0$ wahr ist und sie diese ablehnen. Das ist also gerade ein Fehler erster Art und wir erhalten $\alpha = 0.05$. Damit ist $\varphi = \mathbbm{1}_{[- \lambda_0 \log(\alpha), \infty)} = \mathbbm{1}_{[15,7917, \infty]}$. Wegen $\varphi(X) = \varphi(13) = 0$ publizieren sie ihre Ergebnisse nicht. + \item Es gilt $\mathcal{R}_\lambda = \mathcal F_\lambda^c = (0,\lambda)^c = [\lambda, \infty)$. Die assoziierte Familie von Partitionen ist + \[ + \mathscr H_\lambda^0 \coloneqq \{\tilde \lambda \in \R^+ \colon \lambda \in \mathcal R_{\tilde \lambda}\} + = \{\tilde \lambda \in \R^+ \colon \lambda \geq \tilde \lambda\} = (-\infty, \tilde \lambda] + \] und analog + \[ + \mathscr H_\lambda^1 \coloneqq \{\tilde \lambda \in \R^+ \colon \lambda \in \mathcal F_{\tilde \lambda}\} + = \{\tilde \lambda \in \R^+ \colon \lambda < \tilde \lambda\} = (\tilde \lambda, \infty). + \] + Nach Aufgabe $b$ wissen wir, dass $\varphi_\lambda = \mathbbm{1}_{[-\lambda \log(\alpha)]}$ ein gleichmäßig bester Test zum Niveau $\alpha$ mit Nullhypothese $\mathscr H_\lambda^0$ gegen die Alternative $\mathscr H_\lambda^1$ ist. Nach Satz 12.33 muss damit die assoziierte Bereichsschätzfunktion $B$ ein gleichmäßig bester $(1-\alpha)$-Konfidenzbereich sein. Die assoziierte Bereichsschätzfunktion ist gegeben durch + \begin{align*} + B(x) &= \{\lambda \in \R^+ \colon \varphi_\lambda(x) = 0\}\\ + &= \{\lambda \in \R^+\colon \mathbbm{1}_{[-\lambda\log(\alpha), \infty)} = 0\}\\ + &= \{\lambda \in \R^+\colon x < -\lambda\log(\alpha)\} + \intertext{$\log(\alpha) < 0$} + &= \{\lambda \in \R^+\colon -x/\log(\alpha) < \lambda\}\\ + &= \left(-\frac{x}{\log(\alpha)}, \infty\right) + \end{align*} + \end{enumerate} +\end{aufgabe} \begin{aufgabe}[] \begin{enumerate}[(a)] \item Beh.: $\hat{\theta}_n(x) = (\overline{x}_n, \frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{x}_n)^2)$. @@ -116,5 +157,47 @@ \end{proof} \end{enumerate} \end{aufgabe} - +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Für die Zähldichte $\mathbbm p_{X_i}$ gilt + \[ + \mathbbm p_{X_i} = \begin{cases} + \mathbbm p_{\text{Poi}_\lambda}(0) &| X_i = 0\\ + \sum_{i = 1}^{\infty} \mathbbm p_{\text{Poi}_\lambda}(i) &| X_i = 1 + \end{cases} = \begin{cases} + e^{-\lambda} & |X_i = 0\\ + 1 - e^{-\lambda} & | X_i = 1 + \end{cases} + \] + Für die Produktzähldichte erhalten wir daher + \[ + \prod_{i=1}^n \mathbbm p_{X_i} = (1 - e^{-\lambda})^{\sum_{i = 1}^{n} X_i} (e^{-\lambda})^{n - \sum_{i = 1}^{n} X_i} + = (1 - e^{-\lambda})^{n \overline{X_n}} (e^{-\lambda})^{n - n \overline{X_n}} + %(1 - e^{-\lambda})^{\sum_{i = 1}^{n} X_i} \cdot e^{-n\lambda} \cdot e^{\lambda \sum_{i = 1}^{n} X_i} = (e^{\lambda} - 1)^{\sum_{i = 1}^{n} X_i} \cdot e^{-n\lambda} + \] + Daraus ergibt sich die Likelihoodfunktion + \[ + L(X, \lambda) = (1 - e^{-\lambda})^{n \overline{X_n}} e^{-n\lambda} e^{n \lambda \overline{X_n}} = (e^{\lambda} - 1)^{n \overline{X_n}} \cdot e^{-n\lambda}, + \] + die log-Likelihoodfunktion + \[ + l(X,\lambda) = n \overline{X_n} \log(e^\lambda - 1) - n\lambda, + \] + sowie deren Ableitung + \[ + \frac{\partial}{\partial \lambda} l(X, \lambda) = n \overline{X_n} \frac{e^\lambda}{e^\lambda - 1} - n \overset{!}{=} 0, + \] + von der wir direkt die Nullstellen berechnen um Extremstellen in $L(X, \lambda)$ zu finden. + Daraus folgern wir + \begin{align*} + \overline{X_n} e^{\hat \lambda_n} &= e^{\hat \lambda_n} - 1\\ + 1 - \overline{X_n} &= e^{-\hat \lambda_n}\\ + \hat \lambda_n &= -\log(1 - \overline{X_n}). + \end{align*} + \item Der Schätzer $ \hat \lambda_n$ existiert genau dann nicht, wenn $\overline{X_n} = 1 \Leftrightarrow X_i = 1 \forall 1\le i\le n$. Es gilt + \[ + \mathbbm P(X_1 = \dots = X_n = 1) = \prod_{i=1}^n \mathbbm P_i(X_i = 1) = \prod_{i=1}^n (1 - e^{-\lambda}) = e^{-n\lambda} > 0. + \] + \end{enumerate} +\end{aufgabe} \end{document}