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Add complete theory for exercise 1

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Andreas Tsouchlos
2026-02-12 23:21:15 +01:00
parent 771fa14b20
commit e39ae190f3
1 changed files with 388 additions and 91 deletions
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src/2026-02-13/presentation.tex
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@@ -80,6 +80,20 @@
} }
} }
\newlength{\depthofprodsign}
\setlength{\depthofprodsign}{\depthof{$\prod$}}
\newlength{\totalheightofprodsign}
\newcommand{\nprod}[1][1.4]{
\mathop{
\raisebox
{-#1\depthofprodsign+1\depthofprodsign}
{\scalebox
{#1}
{$\displaystyle\prod$}%
}
}
}
% \tikzstyle{every node}=[font=\small] % \tikzstyle{every node}=[font=\small]
% \captionsetup[sub]{font=small} % \captionsetup[sub]{font=small}
@@ -122,7 +136,8 @@
\begin{itemize} \begin{itemize}
\item Einfache Stichprobe \item Einfache Stichprobe
\begin{gather*} \begin{gather*}
X_1, \ldots, X_N \hspace{2mm}\overbrace{\text{unabhäng und haben X_1, \ldots, X_N
\hspace{2mm}\overbrace{\text{unabhängig und haben
dieselbe Verteilung}}^{\text{``iid.''}} dieselbe Verteilung}}^{\text{``iid.''}}
\hspace*{5mm} \rightarrow\hspace*{5mm} \hspace*{5mm} \rightarrow\hspace*{5mm}
\bm{X} := \bm{X} :=
@@ -157,7 +172,7 @@
X_1 \\ X_1 \\
\vdots \\ \vdots \\
X_N X_N
\end{pmatrix}\sim f_{\bm{X}}$ \end{pmatrix}\sim P_{\bm{X}}$
}; };
\node[right=of model] (x) { \node[right=of model] (x) {
@@ -246,7 +261,7 @@
X_1 \\ X_1 \\
\vdots \\ \vdots \\
X_N X_N
\end{pmatrix}\sim f_{\bm{X}}$ \end{pmatrix}\sim P_{\bm{X}}$
}; };
\draw[ \draw[
@@ -266,7 +281,7 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Punktschätzer I} \frametitle{Punktschätzer}
\vspace*{-10mm} \vspace*{-10mm}
@@ -276,6 +291,113 @@
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\only<1>{
\begin{tikzpicture}
\node[
rectangle,
densely dashed,
draw,
inner sep=5mm,
] (x) {
$
\bm{x} =
\begin{pmatrix}
26{,}2 \\
27{,}8 \\
25{,}7 \\
\vdots
\end{pmatrix}
$
};
\node[
draw opacity=0,
fill opacity=0,
rectangle,
right=of x,
minimum width=5cm, minimum height=2cm,
draw=kit-green, fill=kit-green!20,
line width=1pt,
align=center,
inner sep=3mm
] (est) {Schätzer\\[5mm] $T_N(\bm{x}) =
\displaystyle\frac{1}{N}
\nsum_{i=0}^{N} x_i$};
\node[
draw opacity=0,
fill opacity=0,
above=of est,
rectangle,
densely dashed,
draw,
inner sep=5mm,
] (model) {
$X_i \sim \mathcal{N}(\mu = \vartheta,
\sigma^2 = 1)$
};
\node[right=of est, draw opacity=0, fill
opacity=0] (theta) {$\hat{\vartheta} = 26{,}0$};
\node[below] at (x.south) {Beobachtung};
\node[above, draw opacity=0, fill opacity=0]
at (model.north) {Parametrisiertes Modell};
\end{tikzpicture}
}%
\only<2>{
\begin{tikzpicture}
\node[
rectangle,
densely dashed,
draw,
inner sep=5mm,
] (x) {
$
\bm{x} =
\begin{pmatrix}
26{,}2 \\
27{,}8 \\
25{,}7 \\
\vdots
\end{pmatrix}
$
};
\node[
draw opacity=0,
fill opacity=0,
rectangle,
right=of x,
minimum width=5cm, minimum height=2cm,
draw=kit-green, fill=kit-green!20,
line width=1pt,
align=center,
inner sep=3mm
] (est) {Schätzer\\[5mm] $T_N(\bm{x}) =
\displaystyle\frac{1}{N}
\nsum_{i=0}^{N} x_i$};
\node[
above=of est,
rectangle,
densely dashed,
draw,
inner sep=5mm,
] (model) {
$X_i \sim \mathcal{N}(\mu = \vartheta,
\sigma^2 = 1)$
};
\node[right=of est, draw opacity=0, fill
opacity=0] (theta) {$\hat{\vartheta}
= 26{,}0$};
\node[below] at (x.south) {Beobachtung};
\node[above] at (model.north) {Parametrisiertes Modell};
\end{tikzpicture}
}%
\only<3->{
\begin{tikzpicture} \begin{tikzpicture}
\node[ \node[
rectangle, rectangle,
@@ -302,9 +424,9 @@
line width=1pt, line width=1pt,
align=center, align=center,
inner sep=3mm inner sep=3mm
] (est) {Schätzer\\[5mm] $T(\bm{x}) = ] (est) {Schätzer\\[5mm] $T_N(\bm{x}) =
\displaystyle\frac{1}{N} \displaystyle\frac{1}{N}
\sum_{i=0}^{N} x_i$}; \nsum_{i=0}^{N} x_i$};
\node[ \node[
above=of est, above=of est,
@@ -313,7 +435,8 @@
draw, draw,
inner sep=5mm, inner sep=5mm,
] (model) { ] (model) {
$X_i \sim \mathcal{N}(\mu = \vartheta, \sigma^2 = 1)$ $X_i \sim \mathcal{N}(\mu = \vartheta,
\sigma^2 = 1)$
}; };
\node[right=of est] (theta) {$\hat{\vartheta} \node[right=of est] (theta) {$\hat{\vartheta}
@@ -327,26 +450,113 @@
\draw[-{Latex}, line width=1pt] (model) -- (est); \draw[-{Latex}, line width=1pt] (model) -- (est);
\draw[-{Latex}, line width=1pt] (est) -- (theta); \draw[-{Latex}, line width=1pt] (est) -- (theta);
\end{tikzpicture} \end{tikzpicture}
}
\end{figure} \end{figure}
\pause \pause
\pause
\item Punktschätzer: Rechenvorschrift zur Berechnung von \item Punktschätzer: Rechenvorschrift zur Berechnung von
Parametern aus Beobachtungen \\ Parametern aus Beobachtungen \\
\pause
$\rightarrow$ Schätzer hängen von den Realisierungen ab $\rightarrow$ Schätzer hängen von den Realisierungen ab
und sind damit selbst auch zufällig \\ und sind damit selbst auch zufällig \\
$\rightarrow$ Schätzer haben selbst einen Erwartungswert $\rightarrow$ Schätzer haben einen Erwartungswert und eine Varianz
und eine Varianz
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Punktschätzer II} \frametitle{Likelihood und Log-Likelihood (Diskret)}
\vspace*{-10mm}
\begin{itemize}
\item Maximum Likelihood (ML) Schätzer\\
\begin{minipage}{0.21\textwidth}
\phantom{a}
\end{minipage}
\begin{minipage}{0.16\textwidth}
\centering
\begin{align*}
\hat{\vartheta}_\text{ML}
= \argmax_\vartheta \hspace{2mm} P(\bm{X} = \bm{x}
\vert \vartheta)
\end{align*}
\end{minipage}%
\visible<2->{
\begin{minipage}{0.15\textwidth}
\centering
\begin{align*}
\hspace*{-3mm} = \argmax_\vartheta
\hspace{2mm} L_{\bm{x}} (\vartheta)
\end{align*}
\end{minipage}%
}
\visible<3->{
\begin{minipage}{0.13\textwidth}
\centering
\begin{align*}
\hspace*{-10mm} = \argmax_\vartheta
\hspace{2mm} l_{\bm{x}} (\vartheta)
\end{align*}
\end{minipage}%
}
\begin{figure}[H]
\centering
``Welches $\vartheta$ maximiert die
Wahrscheinlichkeit die beobachtete Realisierung zu bekommen?''
\end{figure}
\pause
\item Likelihoodfunktion
\end{itemize}
\vspace*{5mm}
\begin{minipage}{0.5\textwidth}
\centering
\begin{align*}
L_{\bm{x}}(\vartheta) = P(\bm{X} = \bm{x} \vert
\vartheta) \overset{X_i \text{
iid.}}{=\joinrel=\joinrel=} \nprod_{i=1}^{N}
P(X_i = x_i \vert \vartheta)
\end{align*}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\begin{lightgrayhighlightbox}
\vspace*{-3mm}
Beispiel
\vspace*{-10mm}
\begin{gather*}
X_i \sim \text{\normalfont Binomial} (p = \vartheta, K) \\
L_{\bm{x}}(\vartheta) = P(\bm{X}=\bm{x} \vert \vartheta) =
\nprod_{i=1}^{N}
\binom{K}{x_i}\vartheta^{x_i}(1-\vartheta)^{K-x_i}
\end{gather*}
\vspace*{-10mm}
\end{lightgrayhighlightbox}
\end{minipage}%
\vspace*{5mm}
\begin{itemize}
\pause
\item Log-Likelihoodfunktion
\begin{align*}
l_{\bm{x}}(\vartheta) = \ln \left( L_{\bm{x}}(\vartheta) \right)
\end{align*}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Eigenschaften von Punktschätzern}
\vspace*{-10mm} \vspace*{-10mm}
\begin{itemize} \begin{itemize}
\item Erwartungtreue \item Erwartungtreue
\begin{gather*} \begin{gather*}
E(\hat{\vartheta}) = E\big( T(\bm{X}) \big) = \vartheta E(\hat{\vartheta}) = E\big( T_N(\bm{X}) \big) = \vartheta
\end{gather*} \end{gather*}
\begin{figure}[H] \begin{figure}[H]
@@ -359,7 +569,7 @@
\item Konsistenz \item Konsistenz
\begin{gather*} \begin{gather*}
\lim_{N\rightarrow \infty} P_\vartheta \big( \lvert \lim_{N\rightarrow \infty} P_\vartheta \big( \lvert
T_N - \vartheta \rvert \ge \varepsilon \big) = 0 \hat{\vartheta} - \vartheta \rvert \ge \varepsilon \big) = 0
\end{gather*} \end{gather*}
\begin{figure}[H] \begin{figure}[H]
@@ -371,22 +581,108 @@
\vspace*{10mm} \vspace*{10mm}
\pause \pause
\item Effizienz (für erwartungtreue Schätzer) \item Effizienz (für erwartungtreue Schätzer)
\begin{minipage}{0.68\textwidth}
\begin{gather*} \begin{gather*}
V(\hat{\vartheta}) = \frac{1}{J(\vartheta)}, V(\hat{\vartheta}) = \frac{1}{J(\vartheta)},
\hspace*{5mm} J(\vartheta) = - E\left( \hspace*{5mm} J(\vartheta) = - E\left(
\frac{\partial^2}{\partial \vartheta^2} \frac{\partial^2}{\partial \vartheta^2}
\ln \mleft( f_\vartheta (\bm{X}) \mright) l_{\bm{X}}(\vartheta)
\right) \right)
\end{gather*} \end{gather*}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
``Für jedes N hat der Schätzer jeweils die ``Für jedes fixe N hat der Schätzer jeweils die
kleinstmögliche Varianz'' kleinstmögliche Varianz''
\end{figure} \end{figure}
\end{minipage}%
\begin{minipage}{0.3\textwidth}
\begin{lightgrayhighlightbox}
Cramér-Rao Ungleichung \\
\vspace*{-6mm}
\begin{gather*}
V(\hat{\vartheta}) \le \frac{1}{J(\vartheta)}
\end{gather*}
\vspace*{-10mm}
\end{lightgrayhighlightbox}
\end{minipage}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}
\frametitle{Zusammenfassung}
\vspace*{-10mm}
\begin{columns}
\column{\kitthreecolumns}
% \begin{greenblock}{Einfache Stichprobe}
% \vspace*{-8mm}
% \begin{gather*}
% \bm{X} =
% \begin{pmatrix}
% X_1 \\
% \vdots \\
% X_N
% \end{pmatrix},\hspace{5mm}
% X_1, \ldots, X_N \text{ iid.}
% \end{gather*}
% \vspace*{-3mm}
% \end{greenblock}
\begin{greenblock}{Likelihood und co. (diskret)}
\vspace*{-10mm}
\begin{align*}
\text{Likelihoodfunktion: } &L_{\bm{x}} (\vartheta) = P\left(
\bm{X} = \bm{x}
\vert \vartheta \right) \\[3mm]
\text{Log-Likelihoodfunktion: } &l_{\bm{x}}
(\vartheta) = \ln \left( L_{\bm{x}}
(\vartheta) \right) \\[3mm]
\text{ML-Schätzer: } &\hat{\vartheta}_\text{ML} =
\argmax_\vartheta
\hspace{2mm} l_{\bm{x}} (\vartheta)
\end{align*}
\vspace*{-6mm}
\end{greenblock}
\begin{greenblock}{Eigenschaften von Schätzern}
\vspace*{-10mm}
\begin{align*}
\text{Erwartungtreue: } & E\left( \hat{\vartheta}
\right) = \vartheta \\
\text{Konsistenz: } & \lim_{N\rightarrow \infty}
P\left( \lvert \hat{\vartheta}
- \vartheta \rvert \ge \varepsilon
\right) = 0 \\
\text{Effizienz: } & V(\hat{\vartheta}) =
\frac{1}{J(\vartheta)},\hspace{5mm} J(\vartheta) = - E\left(
\frac{\partial^2}{\partial \vartheta^2}
l_{\bm{x}}(\vartheta) \right)
\end{align*}
\vspace*{-3mm}
\end{greenblock}
\column{\kitthreecolumns}
\begin{greenblock}{Erwartungswert \& Varianz Rechenregeln}
\vspace*{-10mm}
\begin{align*}
E(aX) &= aE(X) \\
E(X + b) &= E(X) + b \\
E(X + Y) &= E(X) + E(Y) \\[5mm]
V(aX) &= a^2V(X) \\
V(X + b) &= E(X) \\
V(X + Y) &= V(X) + V(Y)
\end{align*}
\vspace*{-8mm}
\end{greenblock}
\begin{greenblock}{Tschebyscheff Ungleichung}
\vspace*{-8mm}
\begin{align*}
P\left( \lvert X - E(X) \rvert \ge \varepsilon \right) \le
\frac{V(X)}{\varepsilon^2}
\end{align*}
\vspace*{-6mm}
\end{greenblock}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Aufgabe} \subsection{Aufgabe}
@@ -447,22 +743,22 @@
\begin{align*} \begin{align*}
\hspace*{-77mm} \hspace*{-77mm}
L_{\bm{x}}(\lambda) &= P(\bm{X} = \bm{x} | \lambda) = L_{\bm{x}}(\lambda) &= P(\bm{X} = \bm{x} | \lambda) =
\prod_{i=1}^{N} P(X_i=x_i | \lambda) = \nprod_{i=1}^{N} P(X_i=x_i | \lambda) =
\prod_{i=1}^{N} \frac{\lambda^{x_i}}{x_i!} e^{-\lambda} \nprod_{i=1}^{N} \frac{\lambda^{x_i}}{x_i!} e^{-\lambda}
\end{align*} \end{align*}
\vspace*{-3mm} \vspace*{-3mm}
\pause \pause
\begin{align*} \begin{align*}
l_{\bm{x}}(\lambda) &= \ln \left( l_{\bm{x}}(\lambda) &= \ln \left(
L_{\bm{x}}(\lambda) \right) = \ln \left( L_{\bm{x}}(\lambda) \right) = \ln \left(
\prod_{i=1}^{N} \frac{\lambda^{x_i}}{x_i!} \nprod_{i=1}^{N} \frac{\lambda^{x_i}}{x_i!}
e^{-\lambda} \right) e^{-\lambda} \right)
= =
\sum_{i=1}^{N}\left[\ln \left( e^{-\lambda} \right) + \nsum_{i=1}^{N}\left[\ln \left( e^{-\lambda} \right) +
\ln \left( \lambda^{x_i} \right) \ln \left( \lambda^{x_i} \right)
- \ln \left( x_i! \right)\right] - \ln \left( x_i! \right)\right]
= - N \lambda + \sum_{i=1}^{N} \left[ x_i \ln \left( = - N \lambda + \nsum_{i=1}^{N} \left[ x_i \ln \left(
\lambda \right) - \sum_{n=1}^{x_i} \ln \left( n \lambda \right) - \nsum_{n=1}^{x_i} \ln \left( n
\right) \right] \right) \right]
\end{align*} \end{align*}
\vspace*{5mm} \vspace*{5mm}
@@ -472,18 +768,19 @@
\begin{array}{l} \begin{array}{l}
\displaystyle\frac{\partial \displaystyle\frac{\partial
l_{\bm{x}}(\lambda)}{\lambda} = -N + l_{\bm{x}}(\lambda)}{\lambda} = -N +
\frac{1}{\lambda} \sum_{i=1}^{N} x_i \overset{!}{=} 0 \frac{1}{\lambda} \nsum_{i=1}^{N} x_i \overset{!}{=} 0
\Rightarrow \lambda = \frac{\sum_{i=1}^{N} x_i}{N} \\[7mm] \Rightarrow \lambda = \frac{1}{N} \nsum_{i=1}^{N}
x_i \\[7mm]
\displaystyle\frac{\partial^2 \displaystyle\frac{\partial^2
l_{\bm{x}}(\lambda)}{\partial l_{\bm{x}}(\lambda)}{\partial
\lambda^2} = - \frac{1}{\lambda^2} \sum_{i=1}^{N} x_i < 0 \lambda^2} = - \frac{1}{\lambda^2} \nsum_{i=1}^{N} x_i < 0
\end{array} \end{array}
% tex-fmt: off % tex-fmt: off
\right\} \right\}
% tex-fmt: on % tex-fmt: on
\Rightarrow \hat{\lambda}_\text{ML} = \Rightarrow \hat{\lambda}_\text{ML} =
\argmax_\lambda \hspace{2mm} l_{\bm{x}}(\lambda) = \argmax_\lambda \hspace{2mm} l_{\bm{x}}(\lambda) =
\frac{\sum_{i=1}^{N} x_i}{N} \frac{1}{N} \nsum_{i=1}^{N} x_i
% %
% \hat{\lambda}_\text{ML} = \argmax_\lambda % \hat{\lambda}_\text{ML} = \argmax_\lambda
% \hspace{2mm} \ln \left( l_{\bm{x}} (\lambda) \right) % \hspace{2mm} \ln \left( l_{\bm{x}} (\lambda) \right)
@@ -507,8 +804,8 @@
\pause \pause
\begin{gather*} \begin{gather*}
E(\hat{\lambda}_\text{ML}) = E \left(\frac{1}{N} E(\hat{\lambda}_\text{ML}) = E \left(\frac{1}{N}
\sum_{i=1}^{N} X_i \right) \nsum_{i=1}^{N} X_i \right)
= \frac{1}{N} \sum_{i=1}^{N} E(X_i) = \frac{1}{N} = \frac{1}{N} \nsum_{i=1}^{N} E(X_i) = \frac{1}{N}
\cdot N \lambda = \lambda \cdot N \lambda = \lambda
\hspace{7mm}\Rightarrow\hspace{7mm} \text{Schätzer \hspace{7mm}\Rightarrow\hspace{7mm} \text{Schätzer
ist erwartungstreu} ist erwartungstreu}
@@ -523,7 +820,7 @@
\begin{minipage}{0.16\textwidth} \begin{minipage}{0.16\textwidth}
\begin{gather*} \begin{gather*}
E\left( \lvert \hat{\lambda}_\text{ML} - \lambda E\left( \lvert \hat{\lambda}_\text{ML} - \lambda
\rvert > \varepsilon \rvert \ge \varepsilon
\right) \right)
\end{gather*} \end{gather*}
\end{minipage}% \end{minipage}%
@@ -532,7 +829,7 @@
\begin{gather*} \begin{gather*}
= E\left( \lvert \hat{\lambda}_\text{ML} - = E\left( \lvert \hat{\lambda}_\text{ML} -
E\left(\hat{\lambda}_\text{ML}\right) \rvert E\left(\hat{\lambda}_\text{ML}\right) \rvert
> \varepsilon \ge \varepsilon
\right) \right)
\le \le
\frac{V\left(\hat{\lambda}_\text{ML}\right)}{\varepsilon^2} \frac{V\left(\hat{\lambda}_\text{ML}\right)}{\varepsilon^2}
@@ -542,8 +839,8 @@
\pause \pause
\begin{gather*} \begin{gather*}
V\left(\hat{\lambda}_\text{ML}\right) = V \left( V\left(\hat{\lambda}_\text{ML}\right) = V \left(
\frac{1}{N} \sum_{i=1}^{N} X_i \right) = \frac{1}{N} \nsum_{i=1}^{N} X_i \right) =
\frac{1}{N^2} \sum_{i=1}^{N} V(X_i) = \frac{1}{N^2} \nsum_{i=1}^{N} V(X_i) =
\frac{N\lambda}{N^2} = \frac{\lambda}{N} \frac{N\lambda}{N^2} = \frac{\lambda}{N}
\end{gather*} \end{gather*}
\pause \pause
@@ -564,10 +861,10 @@
\begin{gather*} \begin{gather*}
J\left( \lambda \right) = - E J\left( \lambda \right) = - E
\left( \left(
\frac{\partial^2}{\partial \lambda^2} l_{\bm{x}} \frac{\partial^2}{\partial \lambda^2} l_{\bm{X}}
(\lambda) \right) (\lambda) \right)
= - E \left( \frac{1}{\lambda^2} \sum_{i=1}^{N} X_i \right) = E \left( \frac{1}{\lambda^2} \nsum_{i=1}^{N} X_i \right)
= \frac{1}{\lambda^2} \sum_{i=1}^{N} E\left( X_i = \frac{1}{\lambda^2} \nsum_{i=1}^{N} E\left( X_i
\right) = \frac{N}{\lambda} \right) = \frac{N}{\lambda}
\end{gather*} \end{gather*}
\pause \pause
@@ -786,8 +1083,8 @@
\vspace*{-8mm} \vspace*{-8mm}
\pause \pause
\begin{gather*} \begin{gather*}
\overline{z} = \frac{1}{N} \sum_{i=1}^{N} z_{1,i} = 25 \\ \overline{z} = \frac{1}{N} \nsum_{i=1}^{N} z_{1,i} = 25 \\
s^2 = \frac{1}{N-1} \sum_{i=1}^{N} \left( z_{1,i} - s^2 = \frac{1}{N-1} \nsum_{i=1}^{N} \left( z_{1,i} -
\overline{z} \right)^2 = 4 \overline{z} \right)^2 = 4
\end{gather*} \end{gather*}
\end{minipage}% \end{minipage}%
@@ -811,8 +1108,8 @@
\vspace*{-8mm} \vspace*{-8mm}
\pause \pause
\begin{gather*} \begin{gather*}
\overline{z} = \frac{1}{N} \sum_{i=1}^{N} z_{1,i} = 34{,}875 \\ \overline{z} = \frac{1}{N} \nsum_{i=1}^{N} z_{1,i} = 34{,}875 \\
s^2 = \frac{1}{N-1} \sum_{i=1}^{N} \left( z_{1,i} - s^2 = \frac{1}{N-1} \nsum_{i=1}^{N} \left( z_{1,i} -
\overline{z} \right)^2 \approx 1525{,}84 \overline{z} \right)^2 \approx 1525{,}84
\end{gather*} \end{gather*}
\end{minipage} \end{minipage}