From 665902822406cd590c95584b64af0d922eda1dbb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Sun, 20 Apr 2025 22:08:16 +0200 Subject: [PATCH 1/2] prednaska 15.4.2025 --- parametricka-inference.tex | 28 ++++++++ skripta.pdf | Bin 324474 -> 339968 bytes skripta.tex | 1 + statisticke-funkcionaly.tex | 128 +++++++++++++++++++++++++++++++++++- 4 files changed, 156 insertions(+), 1 deletion(-) create mode 100644 parametricka-inference.tex diff --git a/parametricka-inference.tex b/parametricka-inference.tex new file mode 100644 index 0000000..77c1558 --- /dev/null +++ b/parametricka-inference.tex @@ -0,0 +1,28 @@ +\section{Parametrická inference} + +V této kapitole se budeme věnovat problému odhadování parametru tak, aby získané rozdělení co nejvhodněji pasovalo na experimentální data. Hlavním objektem zkoumání bude rodina parametrických modelů +$$ \mathcal{F} := \{f(\cdot, \vec \theta): \vec \theta \in \vec \Theta \subseteq \R^d\}, $$ +kde $\vec \Theta$ je parametrický prostor a $\vec \theta = (\theta_1, \dots, \theta_d)$ je parametr. + +Je zřejmé, že ne každý model je schopen pokrýt všechna možná rozdělení vyskytující se v přírodě. Musíme proto aproximovat a umět dobře odhadnout, kdy máme ``dost dobrý" odhad. +Budeme se zajímat o odhad nějaké funkce $T(\vec \theta)$. Například pro $X_i \sim N(\mu, \sigma^2)$, pokud naším parametrem zájmu je $\mu$, stačí volit $T(\vec \theta) = \mu$ a $\sigma^2$ se potom nazývá \textit{nežádoucí/rušivý parametr}. + +\begin{example} + Připomeňme si, že náhodná veličina $X$ má rozdělení $\Gamma(a, p)$, jestliže + $$ f_X(x; a, p) = \frac{a^p}{\Gamma(p)} x^{p - 1} \exp\{-ax\}.$$ + kde $a, p > 0$ a + $$ \Gamma(p) = \int_0^\infty y^{p-1}e^{-y} dy. $$ + + Parametrem ve smyslu úvodu je tedy vektor $\vec \theta = (a, p)$. Chceme-li spočítat průměrnou délku života (což je jedna z věcí, k modelování kterých se používá Gamma rozdělení), dostáváme + $$ T(a, p) = \E_{\vec\theta} X = \int_0^\infty \frac{a^px^p}{\Gamma(p)} e^{-ax} dx = \frac{1}{a\Gamma(p)} \int_0^\infty y^p e^{-y}dy = \frac{\Gamma(p + 1)}{a\Gamma(p)} = \frac{p}{a}. $$ +\end{example} + +V dalším textu uvažujme náhodný výběr $X_1, \dots, X_n \overset{IID}\sim F \in \mathcal{F}$. + +\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 $\sigma^2$. +\end{example} + +\hfill \textit{konec 17. přednášky (15.4.2025)} diff --git a/skripta.pdf b/skripta.pdf index fe859fbb062d042114d3ec3f45c0413fe1cc4118..c0276db7d2f69df3aae16035267118cd5a9c2c9d 100644 GIT binary patch delta 46662 zcmZs>LzE_3w5*$~th8<0wr$(CjX!PMW~FW0wr$&YkN0lp+;MudvAT0bd^6(X1lfNN z#XlIh2?YBvE#VQE6u`{E#LmYD&qoSXr$NW zug~9Pl|%qsEnukm_HK6<9^mcxcKPT+P#OK4{Ix%Pz9$Ke50K@+;jJu&eC_dge6C*E z*};{xrq#bcPg;5E@X;C}x8HsLnHl4_yoK3+bOE@t^dDqFVm$xp?V{|Rpy;7@=cHOC z#xuI_-Mj7ayoaBPKl2~=eo-SUhZiS9g-y^jGh>8R#+wwxbLu$ssLYThMbHAtR#`fN zrP%)=*WacI0xa}bqR2EUiir--1iNq~HjHPck1k9{=c}%eJ5=GmuTLj*0Uj!Rf*l#0ju1@B`&DhQeg=N0n682i^(JTK{9={)0Lb-3r!J;XkMBqt>#ocqEi0bh zx+X+JM7l(_Oq%Of-umL#Q?f%gT$LCpG4pGtO3po8SK?Be#X42i$Lgj?L`|kd8OuUP z7A({m|3+65q-OTVC%*@FsjEoo`w{QFiY9p|K`u0&cRKJDhv zIF60qK314oEfkx2-eev5$J$!r!jY~{FxuECZT7x?551)?w_}kpuq668!wp51@o*>8 z#*{nlxR)p1v`tf#Zf?)VN#`ihk^PwrOkM;K)Ve7jwnWIK!g+T{is4`Zz z0N&uYGEf*MGkf@7Zic8x2>Ju->IEBGe7~V%{~S2F=Cohux6}P7QE)GaB?QMmJ?vzw z^@awRn8!Xt4sQ;z0qyu(lKt~W;OpKFsoiVuZ2@&q1AABU8A|tM@;!Ny0do{pt3KSaMNRHYpuKmrZ0SO2doNB2tNv=KCMA0ItVZ;bgzpQZ1|eobHs?0 z1c3&7^=J^7w}l!ej`($p>sV2M`tfW^-yc$Hzd84r)y{^~H0~?c9s^9>^Hv9Y0IqGM zqo;1Jw5iw)&I2lY+uE~`vc+*3XljXiQNh1Q6ny%fQgN^0rifh#oh>mlpp1RXkdo!` z8iwXK8w35dRF|8ERcseSdC?MA$It`q`l6t+$!+c`%dkO~D2s&$0JaFpB%3as4<9VuY!=s$Jc4kH!8ag9Fxg;pGUgnc zrJ^KJ!!v12AIONogL1D@5zQ^|V&|>PFSh=w@0?FKMbse}p3vDwFc?*2G8$li8+ zC1ZgsFv4xQ{$_o_vg_{KMUmnCs0dr2(_pG0peR(kdrulF+?yyoW*`$0u-?r8#%0hL z@7g9_bMiNzA%c;Qxi$cJ(sy+Y-hauo>91afDP~GD-!GBR%#txa*`MvTFdlM6MCRoM z&!;~YAkO$-7(LGhe?gKCLW8Nb07RxVOUj`}Vz!4GWp zu4@KRNeq2-hW>Y1YD(rH03KId1!n0f;-LcFaN^GQQlg&=(h2luTCHmsB|qEhq&ouU+Q_q0OeLtfeSCOF!4(ZOaF50XpFf80+-suVTXM2irsI9?Nd8U-KK*-fQ4ldza3T8pTzLy0^WMP-+($Dn{+^r}&omiLwZ5o)?9x zzXGuNX11_s-&t3?rnl)$8gC`utQ^oIMj zl8#?5&}|uo0Lf7*u<;e)FsW=x!ZI_kYn1kOFGy*SSmJOOCak#`D1r--#zn2zL=Z6I zkNC%iQtaRg8hHj7$GHHrOh3U*%*sCEEws8Wsxq6<(v6vcgJ9ij5|ln=`D4sFLao_2 zs60;*ZXT$3;>YqQe7*B-wUIIml0#s22?ZL8?mpmrfYqu_cm9*&!Y*V!EY&%R=H37_ zVlm;dzh65vA%DgMWIvH=|026{{EZE|;Tl|td;uI!Y%^U6u`L0>20q}T$DjHvHn2Vn&F3FW}>H zvwDH|HAF*|&UB#DU4~Kaye7Bl?Dq6|R+_cDmc+Q;BvdQg-T;>QoOmFPq7Zv;&pD_lnw1(>$T zE(4!r>g1RHtx>}|%Fy)fL=$gHH+eq>1XK~kcN_D3AqI3MRpm?H(Q#K?)}zg48GLD* z+wU@B1zf8sc9N`cj`xQH+eVZWG>VCD>h%~LvQr_z@Nk#Bg^T@ES&$I${CU2d3r7_X zV}vh66wyt5st+p0%0UsGQFTz4uo5;ES5oezf15!2!9nyoAiW$^hVvY%MDqax5D=AJ zhap;eNfeuhnvLm=XQ({g&=O4Lb$BiEt8_!Yfem$$|B4UlUFa2hco8@kItW4(@Ba}j z{7cVC**5Oav**|^{bSMDQ40(UOLityk^V+;ly~~j+h)xDF~vs?B!3>>)~YpIdJu=( zD4Xhq{M*>&@!zknZtgSMT`x5T5GO`Wyq+VY=0Cwdrp`IuoaO`a+6;l|tF1B)-nN`; zd+1xY61P{{D7ZVXYK9+f;Ul@NeI)DSCBUgI7m6w#J3{JGOOY_rwk9?9h@op%;bSw! z+5*8t&Z+0+>-ggZ$d7kWpuaQFHL6rM=8bq21neDsJZrIXY?B*89ZNgEkvd}v#TNqAsV2EOr$^=~JE1CEM5#m% ziMabL)f>4eR;h*7HGmR8$-9UY|Dcj#htrYYu^lA}pmD1gWs&`{WTIaOY0?#)Ce4>o zw&FZ^u~{f-aWgqPHS@W8uimqU)80M7UTyWp&QhaZ2cIbpH_#|h{FYVbLp(7SFm$+k z3G=!_5gyl;^S(HzYzK2*?N5$QQk(LssUa7|kc@J441$l5@K+D8J_?ezTi$2BAl;do z%crK_bIWsyzz;@SU9>I6n&Br#g4;b=v1T-+^GFn4fE%&Vll$31d@?0bo5RQn+?%+@ z1rz0cxH-+a1Xs2}zuN>tOI#`Zq9a}l0ZG@&SS4p>X?!-};)Q+4>JXn2XhbauoQk#F zaJMCiXgxYiibV(*JI8~W#Dr~KFWtEDC=kr1F<}C7;B^a+gH;G`BN`7Vmxa7H*1^hm z=$&P-@9x+n*qL+C!lL5fV+#jVWxVAgu+4c+sYaH65zY4&*IZQ*l~z-Gw;ZN#xXg#i z!M={CtoO|C^bBLHfAc8~8orur54N5+SG<9e64N1kG_eEhFZK@z4#?*VA zy*W9UP5(kj#50ZAK)s{c$O5&sW3ZnGwZ5Yr2N$&Ijs63hgUTxlwvm((;d zz}f)?yUhhWU9UjSPIg*b33NXJ-MaR3D_VB9%=D~H5|f2exG=fo)-=vn=?2f*`-^yt z+C@lcbn-n-=9sND%JrL8WH^?g4Gjf&-D#KH-h5Wn5q54(6?{}&2h-b6YWQ{Q3qNNQ z)#-fJpi^AJvN$Jxt^t^bmTT3W6b+n=+^0WNoa_OB6xE~`Q`D@{f^!DIo2mmUZY^*9 zKDxA9AdHwTxp%>0ZM+S&?m~pvI9_mcK)fxz$$P~!E{&O|yOi8k{Enxocm&r)_zSA} zyC_NHMO7pR}=G&*_%TggFE)Pi`+gu=inr6tFF$7O3MzU8+2-y_1EWf5Ve- z79mSdJ@l`&5&QrWxPX)+T3-R2ZPS2ic<8VHO zYLDsa!W79M7}3JEYLDK`N6zeN!>1zzH|=jw&R|YZzR;KX>sz>e>XB;2s z+PFFz?X4cZ`wSI?MAN1=KxL+(Wc#wLch`D)*P=XUu*k;> z{_N=XQ22}NCv=o&hj+a7p*DbL;5?}Q^_>#?!s4CfTZVd#rxy6V;xVZw`hFYh$V4Qg z_Xfxm5dD7s)^4NS_?6cPA%Fi__R%!#`Wq*YWs>9c?~P``u=mRl;QAdIYbpF*=`rHd z;+5)<#~58eghQlNEE#C;XBa#>oKjLvzd?ES*rgFdSydZU%#UDO0Ft9eAf6Ar5)Ucb#j zx+1C(>g&Y*pfoq(zuW`!@nqMOj~WN6KI+4%BjckJJs);$#h^1y9fO3};a|sDOcc{F z)Pwu+xTrq@m

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+\include{parametricka-inference} \include{ukazkove-pisemky} \end{document} diff --git a/statisticke-funkcionaly.tex b/statisticke-funkcionaly.tex index bb2983a..ba2e91c 100644 --- a/statisticke-funkcionaly.tex +++ b/statisticke-funkcionaly.tex @@ -7,6 +7,132 @@ Nechť $X_1, \dots, X_n$ je IID náhodný výběr z $F$ s rozsahem výběru $n$. $$ \hat F_n(x) = \frac{1}{n}\sum_{i = 1}^n \chi_{\{X_i \leq x\}}. $$ \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 $\frac{1}{n} |\{X_i \leq x\}|$. +Takto definovaná empirická distribuční funkce splňuje všechny vlastnosti normální distribučních funkcí a přiřazuje váhu $\frac{1}{n}$ každému pozorování $X_i$. Taktéž ECDF můžeme definovat jako 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)} + +\begin{theorem}[Bodové vlastnosti ECDF] + \label{thm-pointwise-ecdf} + Pro libovolné pevné $x \in \R$, + \begin{enumerate}[(i)] + \item $\E\left[\hat F_n(x)\right] = F(x)$; + \item $\Var\left[\hat F_n(x)\right] = \frac{F(x)(1 - F(x))}{n}$; + \item $\MSE\left(\hat F_n(x)\right) = \frac{F(x)(1 - F(x))}{n} \to 0$ pro $n \to \infty$; + \item $\hat F_n(X) \overset P \to F(x)$ pro $n \to \infty$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Platí $\E\left[\hat F_n(x)\right] = \E\left[\frac{1}{n}\sum_{i=1}^n \chi_{\{X_i \leq x\}}\right] = \frac{1}{n} \sum_{i=1}^n P[X_i \leq x] = F(x)$. Tím jsme dokázali první vlastnost. + + Dále platí $\Var\left[\hat F_n(x)\right] = \Var\left[\frac{1}{n}\sum_{i=1}^n \chi_{\{X_i \leq x\}}\right] = \frac{1}{n^2} \sum_{i=1}^n \Var[\chi_{\{X_i \leq x\}}] = \frac{1}{n^2} \sum_{i= 1}^n \left[\E \chi^2 - (\E \chi)^2\right] = \frac{1}{n^2} \sum_{i = 1}^n (F(x) - F(x)^2) = \frac{F(x)(1 - F(x))}{n}$, čímž jsme dokázali druhou vlastnost. + + K důkazu třetí rovnosti si uvědomíme, že díky již dokázané vlastnosti (i) je $\bias(\hat F_n(x)) = 0$ a tedy $\MSE\left(\hat F_n(x)\right) = \Var\left[\hat F_n(x)\right]$. + + Nakonec, díky zákonu velkých čísel (Věta \ref{thm-weak-lln}) máme + $$ \hat F_n(x) = \frac{1}{n} \sum_{i = 1}^n \chi_{\{X_i \leq x\}} \overset P \to \E[\chi_{\{X_1 \leq x\}}] = F(x). $$ +\end{proof} + +\begin{definition} + \textit{Funkcionál} je zobrazení $T: \mathcal{F} \to \R$, kde $\mathcal{F}$ je nějaká množina funkcí. +\end{definition} + +\begin{definition} + \textit{Statistický funkcionál} je zobrazení $T$, které přiřadí rozdělení $P_X$ reálné číslo. +\end{definition} + +Můžeme také definovat vektorové funkcionály, stačí obor hodnot nahradit $\R^d$. Uvedeme si několik příkladů statistických funkcionálů. + +\begin{example} + Následující operátory jsou statistické funkcionály: + \begin{itemize} + \item střední hodnota $\mu = \E X = \int x dP_X(x)$; + \item rozptyl $\sigma^2 = \Var X = \int (x - \mu)^2 dP_X(x)$; + \item medián $F^{-1}(1/2) \equiv \inf \{x : P_X((-\infty, x]) > 1/2\}$. + \end{itemize} +\end{example} + +\begin{definition} + Pokud $T(P_X) = \int r(x) dP_X(x)$ pro nějakou měřitelnou funkci $r$, pak $T$ nazýváme \textit{lineární statistický funkcionál}. +\end{definition} + +Motivací této definice je fakt, že takto definovaný funkcionál $T$ je lineární ve svých argumentech, jinými slovy, +$$ T(aP_X + bP_Y + c) = aT(P_X) + bT(P_Y) + c $$ +pro $a, b, c \in \R$. + +Z předchozího příkladu dostaneme, že střední hodnota a rozptyl jsou lineární a medián není (neexistuje vhodná měřitelná funkce $r$). + +\begin{definition} + Nechť $X_1, \dots, X_n$ je náhodný výběr z $F$ s rozsahem výběru $n$, kde + $X_i : (\Omega, \mathcal{A}, P) \to (\R, \mathcal{B}(\R))$ pro $i = 1, \dots, n$. Pak + $$ \hat P_n(B) := \frac{1}{n} \sum_{i=1}^n \chi_{\{X_i \in B\}} \equiv \frac{1}{n} \sum_{i=1}^n \delta_{X_i} (B) $$ + pro $B \in \mathcal{B}(\R)$ se nazývá \textit{empirická pravděpodobnostní míra}. +\end{definition} + +Právě definovaný objekt je \textit{náhodná} pravděpodobnostní míra, které má diskrétní rovnoměrné pravděpodobnostní rozdělení (součet Diracových měr) na náhodných bodech $X_1, \dots, X_n$, kde každý tento bod má váhu $\frac{1}{n}$. + +\begin{definition} + \textit{Plug-in odhad} neznámého parametru $\theta = T(P_X)$ je $\hat \theta_n := T(\hat P_n)$. +\end{definition} + +Myšlenkou definice plug-in odhadu je nahrazení neznámé pravděpodobnostní míry jejím odhadem. + +\begin{example} + Platí $\hat F_n(x) = \hat P_n((-\infty, x])$ pro $x \in \R$. +\end{example} + +\begin{definition} + \textit{Empirický (plug-in) odhad} pro lineární statistický funkcionál $T(P_X) = \int r(x) dP_X(x)$ je + $$ T(\hat P_n) = \int r(x) d\hat P_n(x). $$ +\end{definition} + +\begin{theorem}[Výpočet plug-in odhadu pro lineární statistický funkcionál] + Pro empirický odhad lineárního statistického funkcionálu $T(P_X) = \int r(x) dP_X(x)$ platí + $$ T(\hat P_n) = \frac{1}{n} \sum_{i=1}^n r(X_i). $$ +\end{theorem} + +\begin{proof} + Nechť $\omega \in \Omega$ je dáno. Z definice empirického odhadu lineárního statistického funkcionálu dostáváme + $$ T(\hat P_n)(\omega) = \int_\R r(x) d\hat P_n(\omega)(x) = \int_\R r(x) d\left(\frac{1}{n}\sum_{i=1}^n \delta_{X_i(\omega)}(x)\right) = $$ + $$ \frac{1}{n}\sum_{i=1}^n \int_\R r(x) d\delta_{X_i(\omega)}(x) = \frac{1}{n} \sum_{i=1}^n r(X_i(\omega)). $$ +\end{proof} + +\begin{example} + Spočteme empirickou střední hodnotu. Máme $\mu = T(P_X) = \int x dP_X(x)$ a tedy díky předchozí větě + $$ \hat \mu_n = \int xd\hat P_n(x) = \bar X_n. $$ +\end{example} + +\begin{example} + Spočteme empirický rozptyl. Z definice rozptylu máme + $$ \Var X = \sigma^2 = T(P_X) = \int (x - \mu)^2 dP_X(x) = \int x^2 dP_X(x) - \left(\int x dP_X(x)\right)^2. $$ + Potom + $$ \hat \sigma_n^2 = \int x^2d\hat P_n(x) - \left(\int x d\hat P_n(x) \right)^2 = $$ + $$ \frac{1}{n} \sum_{i=1}^n X_i^2 - \left(\frac{1}{n}\sum_{i=1}^n X_i\right)^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \bar X_n)^2. $$ +\end{example} + +\begin{example} + Spočteme empirickou korelaci. Nechť tedy $Z = [X, Y]^T$ a nechť $\rho = T(P_{[X, Y]^T})$ označuje příslušnou korelaci. + Můžeme psát + $$ T(P_{[X, Y]^T}) = a(T_1(P_{[X, Y]^T}), T_2(P_{[X, Y]^T}) T_3(P_{[X, Y]^T}) T_4(P_{[X, Y]^T}) T_5(P_{[X, Y]^T})),$$ + kde + \begin{align*} + T_1(P_{[X, Y]^T}) &= \int xdP_{[X, Y]^T}(x, y),\\ + T_2(P_{[X, Y]^T}) &= \int ydP_{[X, Y]^T}(x, y),\\ + T_3(P_{[X, Y]^T}) &= \int xydP_{[X, Y]^T}(x, y),\\ + T_4(P_{[X, Y]^T}) &= \int x^2dP_{[X, Y]^T}(x, y),\\ + T_5(P_{[X, Y]^T}) &= \int y^2dP_{[X, Y]^T}(x, y) + \end{align*} + a zároveň $a(t_1, t_2, t_3, t_4, t_5) = \frac{t_3 - t_1 t_2}{\sqrt{(t_4 - t_1^2)(t_5 - t_2^2)}}$. Dosazením se snadno ověří, že tímto jsme opravdu získali vzorec pro daný funkcionál. Nahrazením distribuční funkce jejím empirickým protějškem nakonec dostáváme + $$ \hat \rho = \frac{\sum_i (X_i - \bar X_n)(Y_i - \bar Y_n)}{\sqrt{\sum_i (X_i - \bar X_n)^2 \sum_j (Y_j - \bar Y_n)^2}}.$$ + Tuto veličinu nazýváme \textit{výběrovou korelací}. +\end{example} + + +\begin{definition} + Připomínka: pro $p \in (0, 1)$ definujeme \textit{$p$-tý kvantil} jako + $T(F) = F^{-1}(p) = \inf \{ x : F(x) > p \}$. + + Nyní definujeme + $$ T(\hat F_n) = \hat F_n^{-1}(p) = \inf \{ x : \hat F_n (x) > p \} $$ + a tento objekt nazýváme \textit{$p$-tý výběrový kvantil}. Obdobně definujeme \textit{výběrový medián} jako $\hat F^{-1}_n(1/2)$. Navíc mezikvartilové rozpětí $\tilde T(F) = F^{-1}(3/4) - F^{-1}(1/4)$ lze odhadnout pomocí \textit{výběrového mezikvartilového rozpětí} $\tilde T(\hat F_n) = \hat F_n^{-1}(3/4) - \hat F_n^{-1}(1/4)$. +\end{definition} 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 2/2] 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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