From 4b63402c4256d1008bb36d13126696fb5123f902 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 25 May 2026 16:45:05 +0200 Subject: [PATCH] [PS] Completed tests (examples of likelihood functions, p-value) --- .../ps-jh/parts/07_tests/01_construction.tex | 3 + .../ps/ps-jh/parts/07_tests/02_examples.tex | 52 ++++++++++++++++++ .../ps/ps-jh/parts/07_tests/03_p-value.tex | 22 ++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 438073 -> 446008 bytes .../probability-and-statistics-cheatsheet.tex | 2 + 5 files changed, 79 insertions(+) create mode 100644 semester4/ps/ps-jh/parts/07_tests/02_examples.tex create mode 100644 semester4/ps/ps-jh/parts/07_tests/03_p-value.tex 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 365c519..3fe1c13 100644 --- a/semester4/ps/ps-jh/parts/07_tests/01_construction.tex +++ b/semester4/ps/ps-jh/parts/07_tests/01_construction.tex @@ -20,3 +20,6 @@ Anderer Test $(T', K')$ mit $\P_{\vartheta_0}[T' \in K'] =: \alpha \leq \alpha^* \[ R(x_1, \ldots, x_n) = \frac{\sup_{\vartheta \in \Theta_A} L(x_1, \ldots, x_n; \vartheta)}{\sup_{\vartheta \in \Theta_0} L(x_1, \ldots, x_n; \vartheta)} \] + +% 407 +% TODO: Likelihood-Functions and quotients of common distributions diff --git a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex new file mode 100644 index 0000000..d013dd2 --- /dev/null +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -0,0 +1,52 @@ +\subsection{Beispiele} +\shortexample[Gauss-Test] $\cX_i$ i.i.d. $\sim \cN(\vartheta, \sigma^2)$ unter $\P_\vartheta$ mit bekanntem $\sigma^2$. +Die zu testende Hypothese ist $H_0 : \vartheta = \vartheta_0$, mögliche alternative Hypothesen $H_A$ sind +$\vartheta > \vartheta_0$, $\vartheta < \vartheta_0$ (einseitig) oder $\vartheta \neq \vartheta_0$ (zweiseitig). +Teststatistik ist immer: +\[ + T = \frac{\overline{\cX}_n - \vartheta_0}{\frac{\sigma}{\sqrt{n}}} \sim \cN(0, 1) \qquad \text{unter } \P_{\vartheta_0} +\] +Kritischer Bereich $K$: $(c_>, \8)$, bzw. $(-\8, c_<)$ für einseitig, $(-\8, -c_{\neq}) \cup (c_{\neq}, \8)$ für zweiseitig (verw. falls $|T| > c_{\neq}$). + +\bi{Bestimmung $c$s}: Mit Verteilung von $T$. Bsp: $\alpha = \P_{\vartheta_0}[T > c_>] = 1 - \Phi(c_>)$, folglich $c_> = \Phi^{-1}(1 - \alpha) =: z_{1 - \alpha}$ + + +\shortexample[t-Test] $\cX_i$ i.i.d. $\sim \cN(\mu, \sigma^2)$ unter $\P_{\overrightarrow{\vartheta}}$ +mit $\overrightarrow{\vartheta} := (\mu, \sigma^2)$ (und insbesondere $\sigma^2)$ unbekannt. +Testen wieder $H_0 : \mu = \mu_0$. Teststatistik ist hier: +\[ + T = \frac{\overline{X}_n - \mu_0}{\frac{S}{\sqrt{n}}} \qquad S^2 = \frac{1}{n - 1} \sum_{k = 1}^{n} (\cX_k - \overline{\cX}_n)^2 +\] +Kritischer Bereich $K$: $c_> = t_{n - 1, 1 - \alpha}$, $c_< = t_{n - 1, \alpha} = -c_>$ und $c_{\neq} = t_{n - 1, 1 - 0.5 \alpha}$, +mit $t_{m, \gamma}$ das $\gamma$-Quantil, mit $\P[\cX \leq t_{m, \gamma}] = \gamma$ für $\cX \sim t_m$ + + +\shortremark[Zweistichproben-Tests] $\cX_k$ (i.i.d.) und jeweils unabhängig von allen $\cY_l$ + + +\shortexample[Gepaarter Zweist. m. N.V.] $\cX_i$ i.i.d. $\sim \cN(\mu_\cX, \sigma^2)$ und $\cY_j$ i.i.d. $\sim \cN(\mu_\cY, \sigma^2)$. + +Differenzen $\cZ_k = \cX_k - \cY_k$ unter $\P_\vartheta$ i.i.d. $\sim \cN(\mu_\cX, \mu_\cY, 2\sigma^2)$. +Wenn nicht unabhängig: $\sim \cN(\mu_\cX - \mu_\cY, 2(1 - \rho) \sigma^2)$ mit $\rho \in (-1, 1)$ Korrelation und $\rho = 0$ Unabhängigkeit. + + +\shortexample[Ungepaart] $\cX_i$, $\cY_j$ wie zuvor ($n$ und $m$ je). + +$z$-Test (falls $\sigma^2$ bekannt): +\[ + T = \frac{(\overline{\cX}_n - \overline{\cY}_m) - (\mu_\cX - \mu_\cY)}{\sigma \sqrt{\frac{1}{n} + \frac{1}{m}}} \sim \cN(0, 1) +\] +$\mu_\cX - \mu_\cY$ muss aus $H_0$ bekannt sein. Der kritische Bereich ist wieder Quantile der $\cN(0, 1)$-Verteilung. + +$t$-Test (falls $\sigma^2$ unbekannt): +\[ + T = \frac{(\overline{\cX}_n - \overline{\cY}_m) - (\mu_\cX - \mu_\cY)}{S \sqrt{\frac{1}{n} + \frac{1}{m}}} \sim t_{n + m - 2} +\] +mit $S^2_\cY$ analog zu $S^2_\cX$ ($n$ durch $m$ ersetzt) +\[ + S^2_\cX = \frac{1}{n - 1} \sum_{k = 1}^{n} (\cX_k - \overline{\cX}_n)^2 +\] +Dann: +\[ + S^2 = \frac{1}{m + n - 2}((n - 1)S_\cX^2 + (m - 1)S_\cY^2) +\] diff --git a/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex b/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex new file mode 100644 index 0000000..7ee859a --- /dev/null +++ b/semester4/ps/ps-jh/parts/07_tests/03_p-value.tex @@ -0,0 +1,22 @@ +\subsection{p-Wert} +Für Stichprobe $\cX_i$ ($n$ el.) testen $H_0 : \vartheta = \vartheta_0$ gegen eine $H_A : \vartheta \in \Theta_A$. + +$p$-Wert ist W unter $H_0$ einen mind. so extremen Wert für $T$ zu bekommen wie $T(\omega) = t(x_1, \ldots, x_n)$ (extrem bezieht sich auf $H_A$). +Für bspw. $H_A: \vartheta > \vartheta_0$, dann ist $p\text{-Wert}(\omega) = \P_{\vartheta_0}[T > t_0] \Big|_{t_0 = t(\cX_1(\omega), \ldots)}$ + +\shortdefinition[Geordnete Testsammlung] +$T$ Teststatistik, Familie von Tests $(T, K_t)_{t \geq 0}$ \bi{geordnet bezgl. $T$}, falls $\forall t$ gilt $K_t \subseteq \R$ und $\forall s \leq t$ gilt $K_s \supset K_t$. +Typische Beispiele: +\begin{itemize} + \item $K_t = (t, \8)$ (rechtsseitiger Test) + \item $K_t = (-\8, t)$ (linksseitiger Test) + \item $K_t = (-\8, t) \cup (t, \8)$ (beidseitiger Test) +\end{itemize} + +\shortdefinition[p-Wert] $p\text{-Wert} = G(T) : \Omega \rightarrow [0, 1]$, mit $G(t) = \P_{\vartheta_0}[T \in K_t]$. +Er ist eine Zufallsvariable, hängt direkt von $x_i$ ab. Wiederholen des Tests generiert neuen $p$-Wert. + +Wenn $T$ stetig und $K_t = (t, \8)$, dann gleichverteilt auf $[0, 1]$. + +\inlineintuition Besagt, welche Tests $H_0$ ablehnen würden (alle Tests mit $\alpha > p$ lehnen $H_0$ ab). $H_A$ spielt \textit{keine} Rolle. +Es heisst \textit{NICHT}, dass der $p$-Wert die W \textit{IST}, dass die Hypothese richtig ist. diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index f4cb347d32810f4cb560f8226b95f4a22defd0c6..581904bbe2ad7a10da4f66d160339f2db162ab90 100644 GIT binary patch delta 113692 zcmV(&K;gf+-Ws@^8?a6g0x>X?P!LTdxC8B+?5;hrv&pRIBx`5AyG|ZxsV&i_C=W%& zvw4|c-zs!Blfe)Ze}T@a*47f)WFOjp)+Ne<_pRlq9n-B{^Hbl&r0$ zgtKT4-}m|DGnvh}B*q$sJuif1MCoj%1b{19(f6wp!RE`0?e*zJzS^u|@#d@3i}(06 zU+r+~;;Fj2| z9Pj((W_1nke{wVb_nX~s#cMY6UjsmiVG5Q&kW7{%UhH-st{;8*^)Y^Mi!Bh8MLR`a8K7gL~u$GV91k3qU6WxD?&#PCVmy6khx3dR7x;1XB 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z%Z-`G<;F*lS>~6ek+cisIYOtITs!!O08ZIRY1SWRi@Fw2eFWbDt#6T`{UpZBB88lAsda|l^AEkAThG8eg6 z)7PxZxw%W0(H&Yb5B!mFeJ6P+dbJkds=2+SFhbfTN?OoB(X#@= zotkKMBla7KA)0Zm&})ON9j2eXN2P72u2~k9rPa?hL!HEiticaM%`a@TE+D1L?e913 z9=;eO{K4G9r|><;tNycl#8sw?twBX%mf|DSYF2~3;P)(-NA4|Ig*x(OCI;2- vvZNqC<8H+} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 9f1d5f6..328029c 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -103,6 +103,8 @@ \section{Tests} \input{parts/07_tests/00_basics.tex} \input{parts/07_tests/01_construction.tex} +\input{parts/07_tests/02_examples.tex} +\input{parts/07_tests/03_p-value.tex} % \input{parts/07_tests/}