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zgsL{$W4%VJJbimU$((HQOx7=}j1!q{BWfjIRFEX9t3=YzeqNuvLHrhCqU2tQMkqN~ z?l*c(E%c>$eOh=?=UmOLk)7OcO?qQ_zi;7bY1JY()~f0>n!u;nw$R@YZ7U=}@>iQL z$Ks+@_UVH7qPX4-h-aziRvg(OXV&R#XHC8w+(UK@(j{0U16|t(6zIHpIHs`If7sza zTNwVpD8AeZXrEJ|>lmSPG@|2RVSh7>VF<*tf8UajDHClrE$2bV7J>v2TTCCId$pf= zbaS5zerCWSa(OZ4du*#TAj{w)mHuWDMCZ3qL+Q{ly|a%~AUePq<^6Z%d)I?L5RD2z W2M0L2(8Himh&+s5OibHEhyFi7B=zb5 diff --git a/semester4/ps/ps-rb/parts/07_stats.tex b/semester4/ps/ps-rb/parts/07_stats.tex index 6c9f04c..8148aab 100644 --- a/semester4/ps/ps-rb/parts/07_stats.tex +++ b/semester4/ps/ps-rb/parts/07_stats.tex @@ -177,4 +177,229 @@ $$ \item $\frac{n-1}{\sigma^2}S^2 = \frac{1}{\sigma^2}\sum_{i=1}^n\Bigl( X_i - \bar{X}_n \Bigr)^2 \sim \chi^2_{n-1}$ \item $\bar{X}_n \perp S^2$ \item $\frac{\bar{X}_n-\mu}{\frac{S}{\sqrt{n}}} = \displaystyle\frac{\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}}}{S / \sigma} = \frac{\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}}}{\sqrt{\frac{1}{n-1}\frac{n-1}{\sigma^2}S^2}} \sim t_{n-1}$ -\end{enumerate} \ No newline at end of file +\end{enumerate} + +\newpage + +\subsection{Tests} + +\textbf{Motivation}: Modelliert $\vartheta\in\Theta$ die Daten $\{x_i\}_{i=1}^n$ gut? + +\definition \textbf{Hypothese} $H_0: \Theta_0 \subseteq \Theta$, \textbf{Alternative} $H_A: \Theta_A \subseteq \Theta$ + +\remark Häufig: $\Theta_A = \Theta_0^\text{C} = \Theta \setminus \Theta_0$ + +\definition $\Theta_x$ \textbf{Einfach} $\iffdef |\Theta_x| = 1$\\ +\subtext{Ansonsten: \textit{Zusammengesetzt}} + +\definition \textbf{Test} $(T,K)$\\ +\smalltext{Eine \textit{Entscheidungsregel}, um $H_0$ durch $\{x_i\}_{i=1}^n$ zu bewerten} +\begin{align*} + & \textbf{Teststatistik}\quad & T = t(X_1,\ldots,X_n) \quad t: \R^n\to\R \\ + & \textbf{Kritischer Bereich}\quad & K \subseteq \R +\end{align*} + +\textbf{Entscheidungsregel}: $H_0$ verworfen $\iff T(\omega) \in K$ + +{\footnotesize + \remark Da $T$ zufällig: $\{T\in K\} = \{\omega\in\Omega\sep T(\omega)\in K\}$ und $\P_\vartheta[T\in K]$ ist berechenbar. +} + +\definition \textbf{Fehler 1. Art}: $H_0$ wird abgelehnt, obwohl korrekt\\ +\subtext{$\vartheta\in\Theta_0$ s.d. $T\in K$. $\P_\vartheta[T\in K]$ für $\vartheta\in\Theta_0$ ist W.-keit 1. Art} + +\definition \textbf{Fehler 2. Art}: $H_0$ wird angenommen, obwohl falsch\\ +\subtext{$\vartheta\in\Theta_A$ s.d. $T\notin K$. $\P_\vartheta[T\notin K]$ für $\vartheta\in\Theta_A$ ist W.-keit 2. Art} + +\subsubsection{Statistisches Vorgehen} + +\definition \textbf{Signifikanzniveau} $\alpha$ s.d. $\underset{\vartheta\in\Theta_0}{\sup}\P_\vartheta[T\in K] \leq \alpha$ + +\definition \textbf{Macht} $\beta: \Theta_A \to [0,1] \quad \vartheta \mapsto \beta(\vartheta) = \P_\vartheta[T \in K]$ + +\method \textbf{Asymmetrisches Verfahren}: +\begin{enumerate} + \item $\alpha$ auswählen. (W.-keit Fehler 1. Art auswählen) + \item $\beta$ maximieren. (W.-keit Fehler 2. Art minimieren) +\end{enumerate} + +{\footnotesize + \remark \textbf{Die Wahl von $H_0,H_A$ ist signifikant.} Gemäss Verfahren ist es schwieriger, die Hypothese zu verwerfen als beizubehalten. Man nutzt d.h. die \textit{Negation} der gewünschten Aussage als Hypothese $H_0$. +} + +\newpage + +\subsection{Konstruktion von Tests} + +\subtext{$X_1,\ldots, X_n$ diskrete/stetig unter $\P_{\vartheta_0}$ und $\P_{\vartheta_A}$} + +\definition \textbf{Likelihood Quotient}\\ +\subtext{$x_1,\ldots,x_n \in \R,\quad \vartheta_0\in\Theta_0,\quad \vartheta_A\in\Theta_A$} +$$ + R(x_1,\ldots,x_n;\vartheta_0,\vartheta_A) := \frac{L(x_1,\ldots,x_n;\vartheta_0)}{L(x_1,\ldots,x_n;\vartheta_A)} +$$ + +Intuitiv, wenn $R$ gross ist, ist $\{x_i\}_{i=1}^n$ deutlich wahrscheinlicher in $\P_{\vartheta_0}$ als $\P_{\vartheta_A}$. + +{\footnotesize + \remark $R(x_1,\ldots;\vartheta_0,\vartheta_A) := +\infty \iff L(x_1,\ldots;\vartheta_A)=0$ +} + +\definition \textbf{Likelihood Quotient Test} \smalltext{$c\geq0$} +$$ + T=R(X_1,\ldots,X_n;\vartheta_0,\vartheta_A) \qquad K = (c,\infty] +$$ + +\lemma \textbf{Neyman-Pearson}\\ +\smalltext{$\forall (T',K') \text{ s.d. } \P_{\vartheta_0}[T'\in K'] = \alpha \leq \alpha^*:$} +$$ + \P_{\vartheta_A}[T'\in K'] \leq \P_{\vartheta_A}[T\in K] +$$ +\subtext{$\Theta_0 = \{\vartheta_0\}, \Theta_A = \{\vartheta_A\}, (T,K) \text{ ist L.Q.-Test},\quad \alpha^* = \P_{\vartheta_0}[T\in K]$} + +D.h. jeder solcher Test mit kleinerem $\alpha$ hat ein kleineres $\beta$. + +{\footnotesize + \remark Die Situation $|\Theta_0|=|\Theta_A| = 1$ ist selten. Man wendet trotzdem eine Verallgemeinerung des oberen Prinzips für komplexere Fälle an. +} + +\definition \textbf{Verallgemeinerter Likelihood Quotient} +\begin{align*} + R(x_1,\ldots,x_n) &:= \frac{\underset{\vartheta\in\Theta_0}{\sup} L(x_1,\ldots,x_n;\vartheta)}{\underset{\vartheta\in\Theta_A}{\sup}L(x_1,\ldots,x_n;\vartheta)} \\ + \tilde{R}(x_1,\ldots,x_n) &:= \frac{\underset{\vartheta\in\Theta_0\cup\Theta_A}{\sup} L(x_1,\ldots,x_n;\vartheta)}{\underset{\vartheta\in\Theta_A}{\sup}L(x_1,\ldots,x_n;\vartheta)} +\end{align*} + +\newpage + +\definition \textbf{Gauss-Z-Test}\\ +\smalltext{$X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal N (\vartheta, \sigma^2)$, $\sigma^2$ bekannt} +\begin{align*} + & H_0: & \vartheta = \vartheta_0 \\ + & H_A: & \vartheta > \vartheta_0 \ \lor \ \vartheta < \vartheta_0 \\ + & T: & T=\frac{\bar{X}_n-\vartheta_0}{\frac{\sigma}{\sqrt{n}}}\sim\mathcal N (0,1) \text{ unter } \P_{\vartheta_0} \\ + & K: & K=\begin{cases} + (c_>,\infty) \lor (-\infty, c_<) & \text{Einseitig} \\ + (-\infty, -c_{\neq}) \cup (c_{\neq},\infty) & \text{Beidseitig} + \end{cases} +\end{align*} + +{\footnotesize + \remark Im beidseitigen Fall verwirft man $H_0$ falls $|T|>c_{\neq}$. + + \remark Man bestimmt $c_>,c_<,c_{\neq}$ durch Verteilung $T$ unter $\P_{\vartheta_0}$ +} + +{\scriptsize + \textbf{Beispiel}: Gauss-z-Test mit Quantilen + \begin{align*} + \alpha &= \P_{\vartheta_0}[T\in K] \\ + &= \P_{\vartheta_0}[T > c_>] \\ + &= 1-\P_{\vartheta_0}[T \leq c_>] = \boxed{1-\Phi(c_>)} + \end{align*} + $c_> = \Phi^{-1}(1-\alpha) =: z_{1-\alpha}$ ist das \textit{$(1-\alpha)$-Quantil} von $\mathcal{N}(1,0)$.\\ + Man verwirft für $\vartheta > \vartheta_0$ also $H_0$ falls: + $$ + \frac{1}{n}\sum_{i=1}^nX_i > \vartheta_0 + z_{1-\alpha}\frac{\sigma}{\sqrt{n}} + $$ + Wegen der Symmetrie von $\mathcal N (0,1)$: $c_< = z_\alpha = -z_{1-\alpha}$ und $c_{\neq}=z_{1-\frac{\alpha}{2}}$ +} + +\definition $t$-\textbf{Test}\\ +\smalltext{$X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu,\sigma^2),\quad \vartheta = (\mu,\sigma^2)$} +\begin{align*} + & H_0: & \mu = \mu_0 \\ + & T: & T=\frac{\bar{X}_n-\mu_0}{\frac{S}{\sqrt{n}}} \sim t_{n-1} \text{ unter } \P_{\vartheta_0} \\ + & S: & S^2 = \frac{1}{n-1}\sum_{i=1}^{n}\biggl( X_k-\bar{X}_n \biggr)^2 \\ + & K: & K=\begin{cases} + (c_>,\infty) \lor (-\infty, c_<) & \text{Einseitig} \\ + (-\infty, -c_{\neq}) \cup (c_{\neq},\infty) & \text{Beidseitig} + \end{cases} \\ + & c: & c = \begin{cases} + c_> = t_{n-1,1-\alpha} \\ + c_< = t_{n-1,\alpha} = -t_{n-1,1-\alpha} \\ + c_{\neq} = t_{n-1,1-\frac{\alpha}{2}} + \end{cases} +\end{align*} + +\definition $\gamma$-\textbf{Quantil} $t_{m,\gamma}$ s.d. $\P[X \leq t_{m,\gamma}] = \gamma$ für $X \sim t_m$ + +\newpage + +\definition \textbf{Gepaarter Zweistichproben-Test}\\ +\smalltext{$m=n,\quad \sigma^2$ identisch} +\begin{align*} + X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu_X,\sigma^2) \text{ unter } \P_{\vartheta} \\ + Y_1,\ldots,Y_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mu_Y,\sigma^2) \text{ unter } \P_{\vartheta} +\end{align*} + +{\footnotesize + \remark $(X_k-Y_k) \overset{\text{i.i.d.}}{\sim} \mathcal{N}\Bigl(\mu_X-\mu_Y,2(1-\rho)\sigma^2\Bigr)$\\ +} +\subtext{$\rho \in (-1,1)$ ist Korrelation $X,Y$. Bei Unabhängigkeit: $\rho=0$} + +\definition \textbf{Ungepaarter Zweistichproben-Test} $\iffdef m \neq n$ +$$\ + T=\frac{\bar{X}_n - \bar{Y}_m - (\mu_X-\mu_Y)}{\sigma\sqrt{\frac{1}{n}+\frac{1}{m}}} \sim\mathcal{N}(0,1) +$$ +Bei unbekanntem $\sigma^2$ wird $S^2$ genutzt: +$$ + S^2 = \frac{1}{m+n-2}\biggl( (n-1)S_X^2+(m-1)S_Y^2 \biggr) +$$ + +\subsection{p-Wert} + +\textbf{Motivation}: Bisherige Tests nutzen alle $\mathcal{N}(\mu,\sigma^2)$. + +\smalltext{ + p-Wert Intuitiv: Die W.-keit, unter $H_0$ einen min. so extremen Wert für $T$ zu bekommen wie $T(\omega)$ aus der Stichprobe. +} + +\definition \textbf{Geordnete Testsammlung} $(T,K_t)_{t\geq0}$ +$$ + \forall t: K_t \subseteq \R \quad\land\quad s\ \leq t \implies K_s \supset K_t +$$ +\subtext{z.B. $(T,K_t)_{t\geq0}$ s.d. $K_t=(t,\infty)$ oder $K_t=(-\infty, -t)$} + +\definition \textbf{p-Wert}\\ +\smalltext{$G:\R_+\to[0,1],\quad G(t) = \P_{\vartheta_0}[T\in K_t]$} +$$ + p\text{-Wert } = G(T): \Omega \to [0,1] +$$ +\subtext{$H_0: \vartheta = \vartheta_0$ einfach,$\quad (T,K_t)_{t\geq0}$ geordnet} + +\remark \textbf{Interpretation} +\begin{align*} + \alpha > p &\implies (T, K_t) \text{ lehnt } H_0 \text{ ab} \\ + \alpha \leq p &\implies (T, K_t) \text{ akzeptier } H_0 +\end{align*} + +{\footnotesize + \remark $p$-Wert ist unabhängig von $H_A$ +} + +\newpage + +\subsection{Konfidenzinterval} + +\textbf{Motivation}: $T$ liefert nur ein $\vartheta$, wieso nicht einen Intervall? + +\definition \textbf{Konfidenzbereich} +$$ + C(x_1,\ldots,x_n) \subseteq \Theta \qquad \tilde{C}(X_1,\ldots,X_n) \subseteq \Theta +$$ +\definition \textbf{Konfidenzbereich, Niveau} $1-\alpha$ +$$ + \forall \vartheta \in \Theta:\quad \P\Bigl[ \vartheta \in C(X_1,\ldots,X_n)\Bigr] \geq 1-\alpha +$$ +\definition \textbf{Konfidenzinterval}\\ +\smalltext{$A,B$ sind Zufallsvariablen} +\begin{align*} + & I = [A,B] \text{ s.d. } \P\Bigl[ \vartheta \in [A,B] \Bigr] = \P[A\leq\vartheta\leq B] \geq 1-\alpha \\ + & A = a(X_1,\ldots,X_n) \qquad B = b(X_1,\ldots,X_n) +\end{align*} + +\newpage + +div + +\newpage \ No newline at end of file diff --git a/semester4/ps/ps-rb/parts/08_tables.tex b/semester4/ps/ps-rb/parts/08_tables.tex index d08b11c..daed01c 100644 --- a/semester4/ps/ps-rb/parts/08_tables.tex +++ b/semester4/ps/ps-rb/parts/08_tables.tex @@ -67,6 +67,21 @@ \end{tabular} \end{center} +\subsection{Werte der trigonometrischen Funktionen} + +\footnotesize +\begin{tabular}{c|c|c|c|c|c|c|c|c|c} + $\alpha$ & $0$° & $30$° & $45$° & $60$° & $90$° & $120$° & $150$° & $180$° & $270$° \\ + & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\frac{5\pi}{6}$ & $\pi$ & $\frac{3\pi}{2}$\\ + \hline + $\sin$ & $0$ & $\frac{1}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $0$ & $-1$\\ + \hline + $\cos$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}$ & $0$ & $-\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$ & $-1$ & $0$\\ + \hline + $\tan$ & $0$ & $\frac{\sqrt{3}}{3}$ & $1$ & $\sqrt{3}$ & & $-\sqrt{3}$ & $-\frac{\sqrt{3}}{3}$ & $0$ & \\ +\end{tabular} +\normalsize + %\subsection{Tyler, The Creator} % \begin{center} % \includegraphics[width=0.3\linewidth]{tyler2.png}