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\include{stochasticke-nerovnosti} \include{stochasticke-konvergence} \include{statisticke-uceni} +\include{statisticke-funkcionaly} \include{ukazkove-pisemky} \end{document} diff --git a/statisticke-funkcionaly.tex b/statisticke-funkcionaly.tex new file mode 100644 index 0000000..bb2983a --- /dev/null +++ b/statisticke-funkcionaly.tex @@ -0,0 +1,12 @@ +\section{Statistické funkcionály} + +Nechť $X_1, \dots, X_n$ je IID náhodný výběr z $F$ s rozsahem výběru $n$. Chceme odhadnout $F$ jejím empirickým protějškem. + +\begin{definition}[ECDF] + Pro $x \in \R$ definujeme + $$ \hat F_n(x) = \frac{1}{n}\sum_{i = 1}^n \chi_{\{X_i \leq x\}}. $$ +\end{definition} + +Takto definovaná empirická distribuční funkce splňuje všechny vlastnosti normální distribučních funkcí a přiřazuje váhu $\frac{1}{n}$ každému pozorování $X_i$. Dále budeme používat relativní četnost $X$ menších nebo rovných pevnému $x$, to znamená $\frac{1}{n} |\{X_i \leq x\}|$. + +\hfill \textit{konec 16. přednášky (14.4.2025)} diff --git a/statisticke-uceni.tex b/statisticke-uceni.tex index 8d69ccd..887b6a7 100644 --- a/statisticke-uceni.tex +++ b/statisticke-uceni.tex @@ -11,6 +11,7 @@ V této kapitole se budeme věnovat základům matematické statistiky, což je Obecně si představujeme měřitelná zobrazení $X_1, \dots, X_n$. V praxi však většinou dostaneme pouze reálná čísla $X_i(\omega)$ pro pro jedno konkrétní $\omega \in \Omega$. Možná rozdělení těchto náhodných veličin budeme modelovat pomocí takzvaných parametrických modelů, tedy množin $\mathcal{F}$ rozdělení, jež se dají parametrizovat konečným počtem parametrů. \begin{example}[Normální model] + \label{ex-normal-model} $$ \mathcal{F} = \left\{ f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left\{- \frac{(x - \mu)^2}{2\sigma^2} \right\}, \mu \in \R, \sigma^2 > 0\right\}. $$ Taková data pochází z normálního rozdělení se dvěma parametry $\mu$ a $\sigma^2$. \end{example} @@ -86,7 +87,7 @@ $$ f(x_1, \dots, x_n; \theta) = \prod_{i=1}^n f(x_i; \theta). $$ Z Věty \ref{thm-mse-bias-var} dostáváme, že $\MSE(\hat\theta_n) = \E_\theta[\hat\theta_n - \theta]^2 \to 0$. Z definice $L_2$ konvergence dostáváme, že $\hat\theta_n \overset{L_2}\to \theta$. Zbytek dostáváme z faktu, že $L_2$ konvergence implikuje konvergenci v pravděpodobnosti. \end{proof} -\begin{example} +\begin{example} \label{ex-coin-consistent} Mějme stejnou situaci jako v Příkladu \ref{ex-coin-bernoulli}. Jelikož náš odhad je nestranný ($\E(\hat p_n) = p$) a $\Var(\hat p_n) = \frac{p(1 - p)}{n} \to 0$ pro $n \to \infty$, dostáváme díky Větě \ref{thm-consistence-sufficient-condition}, že $\hat p_n \overset P \to p$. \end{example} @@ -105,3 +106,53 @@ $$ f(x_1, \dots, x_n; \theta) = \prod_{i=1}^n f(x_i; \theta). $$ Tato definice říká, že interval $C_n$ zachytí $\theta$ s pravděpodobností (přibližně) $1 - \alpha$. Tento parametr nazýváme \textit{pokrytí} intervalu spolehlivosti (CI). Interval spolehlivosti je náhodná veličina, i přestože $\theta$ je pevné deterministické. Pro vícerozměrné prostory uvažujeme kouli/elipsoid spolehlivosti (ale toto rozšíření je komplikovanější, protože na $\R^d, d>1$ neexistuje vhodné uspořádání). \hfill \textit{konec 15. přednášky (8.4.2025)} + +\begin{example}[Interval spolehlivosti a normalita] + Mějme stejný setup jako v Příkladu \ref{ex-normal-model}, tedy $X_1, \dots, X_n \overset{IID}\sim N(\mu, \sigma^2)$ a parametr $\mu \in \R$ je \textit{neznámý} a má být \textit{odhadnut} (bodový odhad a konfidenční interval), $\sigma^2 > 0$ je předpokládáno \textit{známé}. + + Nezávislost a stejná rozdělenost nám poskytuje možnost pracovat s mnohorozměrným vektorem $\vec X := (X_1, \dots, X_n)^T \sim ((\mu, \dots, \mu)^T, \sigma^2 I_n)$. + + Uvažujme bodový odhad $\hat \mu_n = \frac{1}{n} \sum_{i=1}^n X_i \equiv \bar X_n$. Tento odhad je konzistentní, což plyne ze zákona velkých čísel \ref{thm-weak-lln}. + Z linearity normálních rozdělení dostáváme $\sqrt{n}(\hat \mu_n - \mu) / \sigma \sim N(0, 1)$. + + Nechť $u_\beta := \Phi^{-1}(\beta)$ je $\beta$-kvantil ($\beta \in (0, 1)$) standardního normálního rozdělení a nechť $Y = \sqrt{n}\frac{\bar X_n - \mu}{\sqrt{\sigma^2}}$. Potom + $$ P[ -u_{1 - \alpha/2} \leq Y \leq u_{1 - \alpha/2}] = P[Y \leq u_{1 - \alpha/2}] - P[Y \leq - u_{1 - \alpha/2}] = $$ + $$ 1 - \frac{\alpha}{2} - \frac{\alpha}{2} = 1 - \alpha$$ + pro všechna $n \in \N$ a $\mu \in \R$. Ve druhé rovnosti jsme použili vlastnost $u_\beta = -u_{1 - \beta}$, která se snadno ověří přímým dosazením do definice distribuční funkce normálního rozdělení. + + Jednoduchými algebraickými úpravami získáme nerovnost pro $\mu$ (zapíšeme to rovnou ve tvaru intervalu): + $$ \mu \in \left(\bar X_n - u_{1 - \alpha / 2} \sqrt{\frac{\sigma^2}{n}}, \bar X_n + u_{1 - \alpha/2} \sqrt{\frac{\sigma^2}{n}}\right). $$ + Tento interval je $1 - \alpha$ interval spolehlivosti pro hodnotu $\mu$. Tedy s pravděpodobnosti $1 - \alpha$ leží hodnota $\mu$ v tomto intervalu. + + Pro výpočet také můžeme použít centrální limitní větu, v tomto případě dostaneme stejný interval spolehlivosti (s poznámkou, že jde o přibližný, tedy asymptotický interval spolehlivosti). +\end{example} + +Zkoumejme délku získaného intervalu spolehlivosti. Z předchozího příkladu máme délku $2u_{1 - \alpha/2}\sqrt{\frac{\sigma^2}{n}}$. Poznamenejme si, že s klesajícím $\alpha$ (povolená tolerance) roste délka intervalu. Taktéž roste délka intervalu s rostoucím rozptylem $\sigma^2$ a klesajícím rozsahem výběru $n$. Mějme danou délku $d$. Kolik pozorování potřebujeme, abychom získali interval spolehlivosti užší než $d$? Vychází +$$ n \geq \floor*{\frac{4u^2_{1 - \alpha/2} \sigma^2}{d^2}} + 1. $$ + +\begin{theorem}[Interval spolehlivosti založený na normalitě] + Předpokládejme, že $\hat \theta_n$ je asymptoticky standardně normální odhad parametru $\theta$ a $\widehat\se(\hat \theta_n)$ je konzistentní odhad $\se(\theta_n)$, tj. $\widehat\se(\hat \theta_n) - \se(\hat \theta_n) \overset P \to 0$. Nechť $u_{1 - \alpha/2}$ je $(1 - \alpha/2)$-kvantil standardního normálního rozdělení a + $$ C_n = \left(\hat \theta_n - u_{1 - \alpha/2}\widehat\se(\hat\theta_n), \hat\theta_n + u_{1 - \alpha/2}\widehat\se(\hat\theta_n)\right).$$ + Pak + $$P_\theta[\theta \in C_n] \overset {n\to\infty} \to 1 - \alpha.$$ +\end{theorem} + +\begin{proof} + Z definice asymptoticky standardně normálního odhadu máme + $$ \frac{\hat \theta_n - \theta}{\se(\hat\theta_n)} \overset D \to N(0, 1) $$ + a máme konzistentní odhad standardní chyby odhadu + $$ \widehat\se(\hat\theta_n H) - \se(\hat\theta_n) \overset P \to 0. $$ + + Ze Slutského věty (Věta \ref{thm-slutsky}) dostáváme $Y := \frac{\hat\theta_n - \theta}{\widehat\se(\hat\theta_n)} \overset D \to N(0, 1)$. Potom již platí + $$ \lim_{n \to \infty} P[-u_{1 - \alpha/2} \leq Y \leq u_{1 - \alpha/2}] = \Phi(u_{1 - \alpha_2}) - \Phi(u_{\alpha/2}) = 1 - \alpha. $$ +\end{proof} + +Neformálně zapisujeme $\theta_n \approx N(\theta, \widehat\se(\hat\theta_n))$. Přibližně platí pro $95\%$-intervaly spolehlivosti $\alpha = 0.05$ a $u_{0.975} \approx 1.96 \approx 2$ vedoucí k explicitnímu intervalu spolehlivosti $\hat\theta_n \pm 2\widehat\se(\hat\theta_n)$. + +\begin{example} + Pokračujeme v Příkladu \ref{ex-coin-consistent}. Již jsme spočítali $\hat p_n \overset P \to p$, $\se(\hat p_n) = \sqrt{p(1 - p)/n}$ a $\widehat\se(\hat p_n) := \sqrt{\hat p_n(1 - \hat p_n) / n}$. Ze Slutského věty pak máme $\widehat\se(\hat p_n) - \se(\hat p_n) \overset P \to 0$. + + Z centrální limitní věty dostáváme, že $\frac{\hat p_n - p}{\se(\hat p_n)} \overset D \to N(0, 1)$. Dále opětovným použitím Slutského věty získáme $\frac{\hat p_n - p}{\widehat\se(\hat p_n)} \overset D \to N(0, 1)$. Tedy z předchozí věty + $$ \hat p_n \pm u_{1 - \alpha/2} \sqrt{\hat p_n(1 - \hat p_n)/n} $$ + je asymptotický (přibližný) $(1 - \alpha)$-interval spolehlivosti pro $p$. +\end{example}