From be34b4715161dc7207e811e028f02a0c55d34e00 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 6 May 2026 13:00:00 +0200 Subject: [PATCH] [PS] Intro to tests --- .../ps/ps-jh/parts/07_tests/00_basics.tex | 32 ++++++++++++++++++ .../ps-jh/parts/07_tests/01_construction.tex | 2 ++ .../probability-and-statistics-cheatsheet.pdf | Bin 428797 -> 434805 bytes .../probability-and-statistics-cheatsheet.tex | 6 ++++ 4 files changed, 40 insertions(+) create mode 100644 semester4/ps/ps-jh/parts/07_tests/00_basics.tex create mode 100644 semester4/ps/ps-jh/parts/07_tests/01_construction.tex diff --git a/semester4/ps/ps-jh/parts/07_tests/00_basics.tex b/semester4/ps/ps-jh/parts/07_tests/00_basics.tex new file mode 100644 index 0000000..a909667 --- /dev/null +++ b/semester4/ps/ps-jh/parts/07_tests/00_basics.tex @@ -0,0 +1,32 @@ +\subsection{Grundbegriffe} +Wiederum Stichprobe $\cX_1, \ldots, \cX_n$, mögliche Modelle durch Familie $(\P_\vartheta)_{\theta \in \Theta}$ beschrieben und $\vartheta$ ein-/mehrdimensional. +Wir haben \textit{Vermutung} wo in $\Theta$ richtiges $\vartheta$ liegt. + +Wir entscheiden zwischen folgenden, mit $\Theta_0 \cap \Theta_A = \emptyset$.: +\begin{itemize} + \item \bi{Hypothese} $\Theta_0 \subseteq \Theta$ (oft: $H_0: \vartheta \in \Theta_0$) + \item \bi{Alternative} $\Theta_A \subseteq \Theta$ (oft: $H_A: \vartheta \in \Theta_A$) +\end{itemize} +Falls keine Alternative spezifiziert, so gilt $\Theta_A = \Theta_0^C = \Theta \backslash \Theta_0$. +Sie heissen \bi{Einfach}, falls $\Theta_0 = \{ \vartheta_0 \}$, resp. $\Theta_A = \{ \vartheta_A \}$, sonst \bi{Zusammengesetzt} + +\shortdefinition[Test] ist ein Paar $(T, K)$ mit +\begin{itemize} + \item \bi{Teststatistik} $T = t(\cX_1, \ldots, \cX_n)$ eine Zufallsvariable mit messbarer Funktion $t: \R^n \rightarrow \R$ + \item \bi{Verwerfungsbereich} $K \subseteq \R$ +\end{itemize} +\shade{gray}{Entscheidungsregel} $H_0$ wird \textit{verworfen}, falls $T(\omega) \in K$, sonst \textit{angenommen}, bzw. nicht verworfen + +\shade{gray}{Fehlerarten} +\bi{Fehler 1. Art} (Hypothese abgelehnt, ist aber richtig), falls $\vartheta \in \Theta_0$, aber $T \in K$, also +$\P_\vartheta[T \in K]$ für $\vartheta \in \Theta_0$ ist W. für F. 1. Art\\ +\bi{Fehler 2. Art} (Hypothese angenommen, ist aber falsch), falls $\vartheta \in \Theta_A$, aber $T \notin K$, also +$\P_\vartheta[T \notin K]$ für $\vartheta \in \Theta_A$ ist W. für F. 2. Art + +\shortremark Entscheidung hängt dabei von \bi{Realisierung} $\omega$ ab. +Da $T$ eine Z.V. ist, können $\P_\vartheta[T \in K]$ in jedem $\P_\vartheta$ betrachten + +\bi{Signifikanzniveau} $\alpha \in (0, 1)$. Ziel: $\underset{\vartheta \in \Theta_0}{\sup} \P_\vartheta[T \in K] \leq \alpha$ +(Fehler 1. Art so klein wie möglich). Typisch: $\alpha = 0.05$ + +\bi{Macht} $\beta: \Theta_A \rightarrow [0, 1]$, mit $\beta(\vartheta) = \P_\vartheta[T \in K]$ soll möglichst gross werden (Fehler 2. Art so klein wie möglich) diff --git a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex new file mode 100644 index 0000000..95bd58c --- /dev/null +++ b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex @@ -0,0 +1,2 @@ +\subsection{Konstruktion von Tests} +% Slide 401 (P24) diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 70379342a0b05f3dae638028facd4a307df051f4..57642e79ebb1ef36dd80026b02d18f8cb8e89192 100644 GIT binary patch delta 207922 zcmV(_K-9nel^XTL8n8_e0y8j|0cHjhlVT7%B#k@TIoVx%VrP?C&q>zKdUu^X&{A8X zO;H|-if8jOzrIyyG?T#)6MrF_ozJq_w-27YeDLKn%GoR@xngSe^6gA(MTBHCMHvx% z_VR4@dQMLkG|RI2lgBT<`}2#FUtWIy;OWZ;e|tcolWaz3RFf<-GnE;lgqWRPK6w30 zHammgzMo};=VtcNw!EB~OcSF8d|%IAJoxcJ<^ZWaTYWAvq5-fGU4Pm+<-(hipa!NS zW($~-G1ZiC7R}-NKEHe>@);9E8%?q21vivPmCqyxa0M;e0j0GKD{Vbn>8%n ze06&99-kJg9S%I>j3_-@up9t#5Z!FfP8MA0`HS6hx4Pb~-eb!zpGjB^N^%K%f^$(s z!*~m3h{@e}-!C_-Ykzo`>G{9k?0zd>Gr9j707^8Kumqgs!i{*b+kLox^yQcD@cj#T zlbl{%#s=k*Pz`&*I1xgYFDrs0s`>NfsvMeSOa+T-7mEtQg(_jlCD9sS7qAT?whgn_ z3rZ2sme;G(>quucNEuAU5JcUw9}@4Wdn>NDCzQ?qaU!{yFMn??iwd(FILHOeUr9v5 zzs+EXZ)bL#2Nn+{AR<17|KK%Z;Gddp;ny!eoWK3I(R|uiboKx?NXZ=bcmEK&;)<{==caxLPn48! zs`tJ7Pf8e0?SG5q!)mCt_pE}a?%hFL!{8o3PkUJ1BRfGipK7A}&+vKmD)e$Od+>Jl z;77a0ja4mI-EJ1(CY?#ZHjNvF4Knx7|9*NoD>pHIz^l#{EQ9U<<+;p=qzt>Xgaxp$ zCM>Q|)y87tlj3~4`Z=S#lfZaL*60KV`mvtW!p>^3pMTbmGxy(S7Bs756d5N-Fo#4*K3xAKT$q+0|v0Y zUNCJsp%L?^>*BK5?3Q?Xx5c(tog>{#oLLUbib&XVX7RM*KRWizQo@-Ck1U|DmYH{C z8ML7Q%zyqER1ffIz-vYduN8E~7|FQQ-kb2Mp}^hto%>HL1gG}3@|kyF>vj&Bec#oj z_5l2j=N0~!G==QV`rG{98X@|U*n+T;RA^z zdDefJ0yr!f@?4Bw+5jWt2~ z;#T5W*VrgoSUJOu%37o9)~JsM4C!{GTH;18*v9n&Ojwaz18YTb9EcOpb50a>0R1^O zR)0=}LcFS**>O!d)d=M-cRdD51x5r1o_aJlf^-%Wk|@OI8xXxxNFv_K08w9kLM}KEe-G ze&AoDkxltcCAIpJCPF(Q;6>FV^&mEc1An_&09{u+Tmv6J;k0Oy=T4(oMP~kb{&j5e zs3e&}j+KL^VqcD;zXj4?w^eNRieyI2g4Io8KVXO^G}9rf^qq$0W`a^P89$=i*}!|I z`2M(E7l|ELyn~>?k1^N{hyjMjU=JVwMr8unuV6*Z{H)lH8V%G`&SQ8#bTHd9DSsD? 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+\section{Tests} +\input{parts/07_tests/00_basics.tex} +\input{parts/07_tests/01_construction.tex} +% \input{parts/07_tests/} + \end{document}