From d2cd0ed9154b0dc426865f1660f260c300d2a65e Mon Sep 17 00:00:00 2001
From: Walmes Zeviani <walmes@ufpr.br>
Date: Mon, 3 Sep 2018 20:30:07 -0300
Subject: [PATCH] Adiciona slides de Jackknife.

---
 _site.yml               |   2 +
 img/swiss-knife.png     | Bin 0 -> 92203 bytes
 slides/07-jackknife.Rnw | 226 ++++++++++++++++++++++++++++++++++++++++
 slides/config/refs.bib  |  34 ++++++
 4 files changed, 262 insertions(+)
 create mode 100644 img/swiss-knife.png
 create mode 100644 slides/07-jackknife.Rnw

diff --git a/_site.yml b/_site.yml
index f06aad3..3473fc7 100644
--- a/_site.yml
+++ b/_site.yml
@@ -41,6 +41,8 @@ navbar:
         href: tutoriais/04-rel-va.html
       - text: "GNA para Normal - Box Muller"
         href: tutoriais/05-box-muller.html
+      - text: "Jackknife"
+        href: slides/07-jackknife.pdf
       - text: "----------"
       - text: "Arquivos complementares"
       - text: "GNA Uniformes (2015)"
diff --git a/img/swiss-knife.png b/img/swiss-knife.png
new file mode 100644
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diff --git a/slides/07-jackknife.Rnw b/slides/07-jackknife.Rnw
new file mode 100644
index 0000000..6190afc
--- /dev/null
+++ b/slides/07-jackknife.Rnw
@@ -0,0 +1,226 @@
+%-----------------------------------------------------------------------
+
+\documentclass[serif, professionalfont, usenames, dvipsnames]{beamer}
+\usepackage[T1]{fontenc}
+
+% ATTENTION: preamble.tex contains all style definitions.
+\input{config/preamble.tex}
+% \usepackage[backend=bibtex, style=authoryear]{biblatex}
+\addbibresource{config/refs.bib}
+
+<<include = FALSE>>=
+source("config/setup.R")
+@
+
+%-----------------------------------------------------------------------
+
+\title{O método Jackknife}
+\subtitle{Fundamentos e aplicações}
+\date{\small{ \Sexpr{sprintf('Atualizado em %s', Sys.Date())}}}
+
+%-----------------------------------------------------------------------
+
+\begin{document}
+
+{\setbeamertemplate{footline}{}
+  \frame{\titlepage} %--------------------------------------------------
+}
+
+\begin{frame}{}
+
+  {\large Justificativas}
+
+  \begin{itemize}
+  \item Métodos computacionalmente intensivos para inferência
+    estatística são usados quando as abordagens tradicionais não são
+    adequadas.
+    \begin{itemize}
+    \item Resultados assintóticos em pequenas amostras.
+    \item Violação de pressupostos.
+    \item Não existência de mecanísmos de inferência específicos.
+    \end{itemize}
+  \item Tais métodos se baseiam em reamostragem e/ou simulação.
+  \item Podem ser aplicados em muitos contextos.
+  \end{itemize}
+
+  {\large Objetivos}
+
+  \begin{itemize}
+  \item Apresentar a ideia principal sobre o método Jackknife.
+  \item Ilustrar propriedades com aplicações.
+  \end{itemize}
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}
+
+  \mytwocolumns{0.49}{0.49}{
+    \begin{itemize}
+    \item O método Jackknife foi proposto por \cite{Quenouille1956}.
+    \item Jackknife refere-se a um canivete suiço, fácil de carregar e de
+      várias utilidades.
+    \item Devido a isso, \cite{Tukey1958} cunhou o termo em Estatística
+      como uma abordagem geral para testar hipóteses e calcular intervalos
+      de confiança.
+    \end{itemize}
+  }{
+    \begin{center}
+      \includegraphics[width=5cm]{../img/swiss-knife.png}
+    \end{center}
+  }
+
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}{Amostras Jackknife}
+
+  % TODO: http://csyue.nccu.edu.tw/ch/Jackknife_Notes.pdf
+
+  As amostras Jackknife são definidas por deixar $k$ observações de fora
+  do conjunto observado $x = \{x_1, x_2, \ldots, x_n\}$ por vez.
+
+  O caso mais prático e comum é quando $k = 1$. Assim a amostra
+  Jackknife $i$ é definida como
+  \begin{equation}
+    x_{(i)} = \{x_1, x_2, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n\},
+  \end{equation}
+  com $i = 1, 2, \ldots, n$.
+
+  \begin{itemize}
+  \item O tamanho de cada amostra Jackknife é $m = n - k$.
+  \item O número de amostras distintas é $\binom{n}{k}$.
+  \item No caso de $k = 1$, denota-se por $\{x_{(i)}\}$ com $i=1, \ldots, n$.
+  \item As amostras são obtidas de forma sistemática, portanto, trata-se
+    de uma abordagem determinística.
+  \end{itemize}
+
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}{A intuição do método}
+
+  A ideia é fundamentada no estimador da média
+  \begin{equation}
+    \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i.
+  \end{equation}
+
+  A média com a $j$-ésima observação removida é calculada por
+  \begin{equation}
+    \bar{X}_{-j} = \frac{1}{n - 1} \left( \sum_{i=1}^{n} X_i \right) - X_j.
+  \end{equation}
+
+  Combinando as expressões anteriores, pode-se determinar o valor de $X_j$
+  por
+  \begin{equation}
+    X_j = n\bar{X} - (n - 1) \bar{X}_{-j}.
+  \end{equation}
+
+  Essa expressão não tem valor prático para o caso da média, porém tem
+  utilidade para outras estatísticas.
+
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}{Definição}
+
+  Suponha que $\theta$ seja um parâmetro a ser estimado a partir de uma
+  função dos dados
+  \begin{equation}
+    \hat{\theta} = f(X_1, X_2, \ldots, X_n).
+  \end{equation}
+
+  A estimativa em cada amostra Jackknife é aquela obtida deixando $k$
+  observações de fora por vez. No caso de $k=1$, é definida por
+  \begin{equation}
+    \hat{\theta}_{-i} = f(X_1, X_2, \ldots, X_{i-1}, X_{i+1}, \ldots, X_n) =  f(X_{(i)}),
+  \end{equation}
+  é chamada de estimativa parcial.
+
+  A quantidade
+  \begin{equation}
+    \theta_{(i)} = n \hat{\theta} - (n - 1) \hat{\theta}_{-i}
+  \end{equation}
+  é denominada de \hi{pseudo-valor} e se baseia nas diferenças
+  ponderadas da estimativa com todas as observações ($\hat{\theta}$) e
+  na \hi{estimativa parcial}, ou seja, aquela sem a $i$-ésima observação
+  ($\hat{\theta}_{-i}$).
+
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}{Estimativa pontual e variância}
+
+  O estimador pontual de Jackknife é definido por
+  \begin{equation}
+    \tilde{\theta} = \frac{1}{n} \sum_{i = 1}^{n} \theta_{(i)},
+  \end{equation}
+  ou seja, é \hi{a média dos pseudo-valores}.
+
+  ATENÇÃO: Os valores $\hat{\theta}$ e $\tilde{\theta}$ não são
+  necessariamente iguais nos casos gerais.
+
+  Se for assumido que os valores $\theta_{(i)}$, $i = 1, \ldots, n$, são
+  independentes, a variância do estimador de Jackknife e dada por
+
+  \begin{equation}
+    \text{Var}(\tilde{\theta}) = \frac{S_{\theta}^2}{n},
+    \quad S^2 = \frac{1}{n - 1}
+    \sum_{i = 1}^n (\theta_{(i)} - \tilde{\theta})^2.
+  \end{equation}
+
+\end{frame}
+
+%-----------------------------------------------------------------------
+\begin{frame}{Detalhes}
+
+  De acordo com \cite{efron2016computerage}
+
+  \begin{itemize}
+  \item É um \hi{procedimento não paramétrico} pois nenhuma suposição é
+    feita sobre a distribuição dos dados.
+  \item É facilmente automatizável.  Um único algoritmo pode ser escrito
+    tendo como argumentos a amostra e a estatística de interesse $f(.)$.
+  \item É determinístico, portanto, toda execução do procedimento irá
+    fornecer os meus resultados.
+  \item Existe a suposição implicita de comportamento suave da função
+    $f$ em relação a cada elemento da amostra Jackknife.
+  \item O erro-padrão de Jackknife é viciado para estimar o verdadeiro
+    erro padrão, pois os pseudo-valores não são independentes.
+  \end{itemize}
+\end{frame}
+
+%-----------------------------------------------------------------------
+{
+  \usebackgroundtemplate{\includegraphics[height=\paperheight, width=\paperwidth]{../img/looking-ahead.jpg}}
+  % \setbeamersize{text margin left=30mm}
+
+  \begin{frame}[b]{}
+
+    \hspace*{0.5\linewidth}
+    \begin{minipage}[t]{0.5\linewidth}
+
+      \hi{Próxima aula}
+      \begin{itemize}
+      \item Implementação e aplicações de Jackknife.
+      \end{itemize}
+
+      \hi{Avisos}
+      \begin{itemize}
+      \item Sabatina estará disponível a partir de Qua.
+      \end{itemize}
+
+      \vspace{3em}
+    \end{minipage}
+
+\end{frame}
+}
+
+%-----------------------------------------------------------------------
+\begin{frame}[t, fragile, allowframebreaks]
+  \frametitle{Referências bibliográficas}
+
+  \printbibliography[heading=none]
+\end{frame}
+
+%-----------------------------------------------------------------------
+\end{document}
diff --git a/slides/config/refs.bib b/slides/config/refs.bib
index 4f43e63..d44bd0f 100644
--- a/slides/config/refs.bib
+++ b/slides/config/refs.bib
@@ -15,3 +15,37 @@
   year={2013},
   publisher={Editora UFLA}
 }
+
+@article{Quenouille1956,
+  doi = {10.2307/2332914},
+  url = {https://doi.org/10.2307/2332914},
+  year  = {1956},
+  month = {dec},
+  publisher = {{JSTOR}},
+  volume = {43},
+  number = {3/4},
+  pages = {353},
+  author = {M. H. Quenouille},
+  title = {Notes on Bias in Estimation},
+  journal = {Biometrika}
+}
+
+@article{Tukey1958,
+  doi = {10.1214/aoms/1177706647},
+  year  = {1958},
+  volume = {2},
+  number = {29},
+  pages = {614},
+  author = {John W. Tukey},
+  title = {Bias and confidence in not quite large samples (abstract)},
+  journal = {The Annals of Mathematical Statistics}
+}
+
+@Book{efron2016computerage,
+  Title                    = {Computer Age Statistical Inference: Algorithms, Evidence, and Data Science (Institute of Mathematical Statistics Monographs)},
+  Author                   = {Bradley Efron and Trevor Hastie},
+  Publisher                = {Cambridge University Press},
+  Year                     = {2016},
+
+  Url                      = {https://www.amazon.com/Computer-Age-Statistical-Inference-Mathematical-ebook/dp/B01L27MR64?SubscriptionId=0JYN1NVW651KCA56C102&tag=techkie-20&linkCode=xm2&camp=2025&creative=165953&creativeASIN=B01L27MR64}
+}
-- 
GitLab