From e92f8384de63da7020eb56b870a885326789c8bd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Tue, 22 Apr 2025 12:12:51 +0200 Subject: [PATCH] prednaska 22.4.2025 --- parametricka-inference.tex | 84 ++++++++++++++++++++++++++++++++++++- skripta.pdf | Bin 339968 -> 350662 bytes 2 files changed, 83 insertions(+), 1 deletion(-) diff --git a/parametricka-inference.tex b/parametricka-inference.tex index 77c1558..30b0421 100644 --- a/parametricka-inference.tex +++ b/parametricka-inference.tex @@ -22,7 +22,89 @@ V dalším textu uvažujme náhodný výběr $X_1, \dots, X_n \overset{IID}\sim \begin{example} Uvažujme $\mathcal{F} = \{F(\mu): \E_{F(\mu)} = \mu \land |\mu| < \infty$ rodinu modelů s konečnou střední hodnotou. Potom $\bar X_n$ je konzistentní a nestranný odhad $\mu$ a $X_1$ je nestranný, ale ne konzistentní odhad $\mu$. - Dále uvažujme $\mathcal{F} = \{F(\sigma^2): \Var_{F(\sigma^2)} = \sigma^2 < \infty\}$ rodinu modelů s konečným rozptylem. Potom $\hat\sigma_n^2 = n^{-1}\sum_{i=1}^n (X_i - \bar X_n)^2$ je konzistentní, ale ne nestranný odhad $\sigma^2$ a $S_n^2 = (n - 1)^{-1} \sum_{i=1}^n (X_i - \bar X_n)^2)$ je konzistentní a nestranný odhad $\sigma^2$. + Dále uvažujme $\mathcal{F} = \{F(\sigma^2): \Var_{F(\sigma^2)} = \sigma^2 < \infty\}$ rodinu modelů s konečným rozptylem. Potom $\hat\sigma_n^2 = n^{-1}\sum_{i=1}^n (X_i - \bar X_n)^2$ je konzistentní, ale ne nestranný odhad $\sigma^2$ a $S_n^2 = (n - 1)^{-1} \sum_{i=1}^n (X_i - \bar X_n)^2$ je konzistentní a nestranný odhad $\sigma^2$. \end{example} \hfill \textit{konec 17. přednášky (15.4.2025)} + +\begin{example} + Nechť $\mathcal{F} = \{Po(\lambda), \lambda > 0\}$ a $\theta = P[X_i = 0] = e^{-\lambda}$. Potom $\hat \theta_n = n^{-1} \sum_{i=1}^n \chi_{\{X_i = 0\}}$ (relativní četnost nul v původních datech) je konzistentní a nestranný odhad $\lambda$. Zároveň také $\tilde \theta_n = \left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i}$ je konzistentní a nestranný. + + Ukážeme si, že $\hat \theta_n$ je nestranný. Chceme dokázat, že $\E \hat\theta_n = \theta$. Můžeme psát + $$ \E\hat\theta_n = \E\left[\frac{1}{n} \sum_{i=1}^n \chi_{\{X_i = 0\}}\right] = \frac{1}{n}\sum_{i=1}^n P[X_i = 0] = \theta. $$ + Obdobně ukážeme konzistenci tohoto odhadu, tedy, že $\hat\theta_n \to \theta$. Díky slabému zákonu velkých čísel (Věta \ref{thm-weak-lln}): + $$ \hat\theta_n \overset P\to \E[\chi_{\{X_1 = 0\}}] = P[X_1 = 0] = \theta. $$ + + Dále pro odhad $\tilde \theta_n$ můžeme psát (používáme označení $\sum_{i=1}^n X_i = Y$) + $$\E \tilde\theta_n = \E\left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i} = \sum_{y=1}^\infty \left(\frac{n-1}{n}\right)^y \frac{(n\lambda)^y}{y!}e^{-n\lambda} = $$ + $$ e^{-n\lambda}\sum_{y=0}^\infty \frac{[(n-1)\lambda]^y}{y!} = e^{-\lambda},$$ + kde první rovnost plyne z toho, že součet $n$ IID poissonovských náhodných veličin s parametrem $\lambda$ je opět poissonovská náhodná veličina s parametrem $n\lambda$. K důkazu konzistence zlogaritmujeme náš odhad, dostaneme + $$ \log \tilde\theta_n = \left(\sum_{i=1}^n X_i\right)\log\frac{n -1}{n} = \bar X_n \log \left(1 - \frac{1}n\right)^n \overset P \to - \lambda. $$ + Limitní přechod jsme získali díky Slutského větě (Věta \ref{thm-slutsky}). Dále z věty o spojité transformaci (Věta \ref{thm-continuous-mapping}) aplikované na funkci $t(x) = e^x$ dostáváme, že $\tilde \theta_n \overset P \to \theta$. + + Ve speciálním případě $\theta = e^{-2\lambda}$ jediný nestranný odhad je $(-1)^{X_1}$, který ale nikdy nedosáhne přípustné hodnoty $e^{-2\lambda}$. + + Skutečně, nechť existuje nestranný odhad parametru $\theta \in (0, 1)$. Označme ho $\hat\theta_n = T(X_1, \dots, X_n)$. Z definice nestrannosti musí platit, že $\E T(X_1, \dots, X_n) = \theta$. Potom platí + $$\theta = \E \sum_{x=0}^\infty T(x) \frac{\lambda^x}{x!}e^{-\lambda} \overset{\text{předpoklad}} = e^{-2\lambda}. $$ + Z toho však plyne, že + $$\sum_{x=0}^\infty T(x)\frac{\lambda^x}{x!} = e^{-\lambda} = \sum_{x=0}^\infty \frac{(-\lambda)^x}{x!},$$ + a tedy $T(x) = (-1)^x$ (rovnost mocninných řad). +\end{example} + +Dále se budeme věnovat takzvané momentové metodě. Jedná se o univerzální techniku získání odhadů, která však ale nemusí poskytnout ten nejlepší možný odhad. Často ji například využijeme jako startovací bod pro další iterativní numerické metody. + +\begin{definition} + Definujeme \textit{$k$-tý necentrální moment} jako + $$\mu_k' \equiv \mu_k'(\vec \theta) = \E_{\vec \theta} X^k = \int x^k f_X(x, \vec\theta) dP_X$$ + a \textit{$k$-tý výběrový moment} jako + $$ \hat\mu_k' = \frac{1}{n} \sum_{i=1}^n X_i^k. $$ +\end{definition} + +\begin{definition} + \textit{Odhad metodou momentů $\hat{\vec\theta}_n$} je definován jako hodnota $\vec\theta$, pro kterou platí + $$\mu_1'(\hat{\vec\theta}_n) = \hat\mu_1', \dots, \mu_d'(\hat{\vec\theta}_n) = \hat\mu_d'.$$ +\end{definition} + +Alternativně bychom mohli použít centrované $k$-té momenty spolu s jejich empirickými protějšky. + +\begin{example} + Nechť $X_1, \dots, X_n \overset{IID}\sim Be(p)$. Pak $\mu_1' = \E_pX_1 = p$ a $\hat\mu_1' = \bar X_n$. Rovnost těchto dvou hodnot nám dává odhad + $$ \hat p_n = \frac{1}{n} \sum_{i=1}^n X_i = \bar X_n $$ + a je to opět stejný plug-in odhad. +\end{example} + +\begin{example} + Nechť $X_1, \dots, X_n \overset{IID}\sim N(\mu, \sigma^2)$. Pak $\mu_1' = \E_{(\mu, \sigma^2)} X_1 = \mu$ a $\mu_2' = \E_{(\mu, \sigma^2)}X_1^2 = \Var_{(\mu, sigma^2)} X_1 + (\E_{(\mu, \sigma^2)} X_1)^2 = \sigma^2 + \mu^2$. Musíme vyřešit soustavu rovnic + \begin{align*} + \hat\mu_n &= \frac{1}{n} \sum_{i=1}^n X_i;\\ + \hat\mu_n^2 + \hat\sigma_n^2 &= \frac{1}{n}\sum_{i=1}^n X_i^2. + \end{align*} + + Řešením této soustavy je $\hat\mu_n = \bar X_n$ a $\hat\sigma_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar X_n)^2$. +\end{example} + +Je důležité si poznamenat, že někdy si nevystačíme s prvními $d$ momenty, například, pokud je naše teoretické rozdělení symetrické okolo nuly. + +\begin{example} + Nechť $X_1, \dots, X_n \overset{IID}\sim \Gamma(a, p)$, kde + $$ f_X(x; a, p) = \frac{a^p}{\Gamma(p)}x^{p-1}e^{-ax}, x > 0, $$ + kde $a, p > 0$ a $\Gamma(p) = \int_0^\infty y^{p-1}e^{-y}dy$ je Gamma funkce. Pak, odhady momentovou metodou + $$ \hat a_n = \frac{\bar X_n}{\hat\sigma_n^2} \quad\text{a}\quad\hat p_n = \frac{\bar X_n^2}{\hat\sigma_n^2}$$ + jsou konzistentní a AN. + + Skutečně, máme + \begin{align*} + \mu_1'(a, p) &= \E_{(a, p)}X_1 = \frac{p}{a},\\ + \mu_2'(a, p) &= \E_{(a, p)}X_1^2 = \int_0^\infty x^2 \frac{a^p}{\Gamma(p)}x^{p-1}e^{-ax} = \frac{a^p\Gamma(p + 2)}{\Gamma(p)a^{p+2}} = \frac{(p+1)p}{a^2}. + \end{align*} + + Stačí tedy vyřešit soustavu + \begin{align*} + \hat\mu_1' &= \frac{1}{n}\sum_{i=1}^n X_i = \mu_1'(\hat a, \hat p) = \frac{\hat p}{\hat a};\\ + \hat\mu_2' &= \frac{1}{n}\sum_{i=1}^n X_i^2 = \mu_2'(\hat a, \hat p) = \frac{(\hat p + 1)\hat p}{\hat a^2};\\ + \end{align*} + + Jejím řešením jsou právě výše uvedené odhady. +\end{example} + +\hfill \textit{konec 18. přednášky (22.4.2025)} diff --git a/skripta.pdf b/skripta.pdf index c0276db7d2f69df3aae16035267118cd5a9c2c9d..e0fbc1fa7b2d32573acdbcb78330b07033588383 100644 GIT binary patch delta 55705 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