From 033e3b9e4f292006e35b9a795e3cefd1d4b197be Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 6 May 2026 14:07:55 +0200 Subject: [PATCH] [PS] Basics of constructing tests --- .../ps-jh/parts/07_tests/01_construction.tex | 20 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 434805 -> 437699 bytes 2 files changed, 20 insertions(+) diff --git a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex index 95bd58c..365c519 100644 --- a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex +++ b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex @@ -1,2 +1,22 @@ \subsection{Konstruktion von Tests} % Slide 401 (P24) +$\cX_i$ diskret oder gemeinsam stetig unter $\P_{\vartheta_0}$ und $\P_{\vartheta_A}$. Sei $L(x_1, \ldots, x_n; \vartheta)$ die Likelihood-Funktion + +\shortdefinition[Likelihood-Quotient] Für $\vartheta_0 \in \Theta_0$, $\vartheta_A \in \Theta_A$ und $x_i \in \R$: +$R(x_1, \ldots, x_n; \vartheta_0, \vartheta_A) = \frac{L(x_1, \ldots, x_n; \vartheta_A)}{L(x_1, \ldots, x_n; \vartheta_0)}$.\\ +$R(x_1, \ldots; \vartheta_0, \vartheta_A) = +\8$ wenn $L(x_1, \ldots; \vartheta_0) = 0$. +Wenn $R$ gross ist, sind Beobachtungen in $\P_{\vartheta_A}$ deutlich wahrscheinlicher als in $\P_{\vartheta_0}$. + +% TODO: More places with highlighting +\shortdefinition[Likelihood-Quotienten-Test] Sei $c \geq 0$. Der LQT mit param $c$ ist Test $(T, K)$ mit $T = R(\cX_1, \ldots, \cX_n; \vartheta_0, \vartheta_A)$ +und $K = (c, \8)$. $H_0$ wird verworfen wenn $R$ gross wird. + +\shorttheorem[Neyman-Pearson] Seien $\Theta_0, \Theta_A$ \textit{einfach} und $(T, K)$ ein LQT mit $c$ und sig.-niv. $\alpha^* := \P_{\vartheta_0}[T \in K]$. +Anderer Test $(T', K')$ mit $\P_{\vartheta_0}[T' \in K'] =: \alpha \leq \alpha^*$, so gilt auch $\P_{\vartheta_A}[T' \in K'] \leq \P_{\vartheta_A}[T \in K]$. + +\inlineintuition Jeder Test mit kleinerem Signifikanzniveau hat auch kleinere Macht + +\shortdefinition[Verallgemeinerte LQ] ist gegeben durch +\[ + R(x_1, \ldots, x_n) = \frac{\sup_{\vartheta \in \Theta_A} L(x_1, \ldots, x_n; \vartheta)}{\sup_{\vartheta \in \Theta_0} L(x_1, \ldots, x_n; \vartheta)} +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 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