$$
\begin{align}
\gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\
&= \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\
-&=
-\begin{cases}
-\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$}
+&= \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0}
\tag{22}\\
-\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else}
-\end{cases}.
+&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else}
\tag{23}
\end{align}
$$
+
The quantity \( \vec{X} \) is asymptotic uncorrelated by assumption, \( \vec{X}_k \) is also asymptotic uncorrelated. Let's turn our attention to the variance of the sample mean \( V(\overline{X}) \). diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html index f1b10c5e7..cb8f3afa2 100644 --- a/doc/pub/Regression/html/Regression-solarized.html +++ b/doc/pub/Regression/html/Regression-solarized.html @@ -2969,12 +2969,9 @@ $$ \begin{align} \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\ &= \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\ -&= -\begin{cases} -\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} +&= \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} \label{_auto12}\\ -\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else} -\end{cases}. +&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else} \label{_auto13} \end{align} $$ diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index 6e5b47866..8cc2430b9 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -2974,12 +2974,9 @@ $$ \begin{align} \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\ &= \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\ -&= -\begin{cases} -\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} +&= \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} \label{_auto12}\\ -\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else} -\end{cases}. +&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else} \label{_auto13} \end{align} $$ diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb index d0374d81e..65af1be9d 100644 --- a/doc/pub/Regression/ipynb/Regression.ipynb +++ b/doc/pub/Regression/ipynb/Regression.ipynb @@ -3837,9 +3837,7 @@ "\n", "$$\n", "\\begin{equation} \n", - "= \n", - "\\begin{cases}\n", - "\\frac{1}{2}\\gamma_{k}(2h) + \\frac{1}{2}\\gamma_k(2h+1) \\qquad\\qquad\\quad \\ \\ \\text{if $h = 0$} \n", + "= \\frac{1}{2}\\gamma_{k}(2h) + \\frac{1}{2}\\gamma_k(2h+1) \\hspace{0.1cm} \\mathrm{h = 0} \n", "\\label{_auto12} \\tag{22}\n", "\\end{equation}\n", "$$" @@ -3854,8 +3852,7 @@ "\n", "$$\n", "\\begin{equation} \n", - "\\frac{1}{4}\\gamma_k(2h-1) + \\frac{1}{2}\\gamma_k(2h) + \\frac{1}{4}\\gamma_k(2h+1) \\quad \\text{else}\n", - "\\end{cases}.\n", + "=\\frac{1}{4}\\gamma_k(2h-1) + \\frac{1}{2}\\gamma_k(2h) + \\frac{1}{4}\\gamma_k(2h+1) \\quad \\mathrm{else}\n", "\\label{_auto13} \\tag{23}\n", "\\end{equation}\n", "$$" diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz index 0e46aa69b..d69711e82 100644 Binary files a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz and b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz differ diff --git a/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf b/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf index 7357f3649..52f1686f3 100644 Binary files a/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf and b/doc/pub/Regression/pdf/Regression-beamer-handouts2x3.pdf differ diff --git a/doc/pub/Regression/pdf/Regression-beamer.pdf b/doc/pub/Regression/pdf/Regression-beamer.pdf index fd2fb960f..98855d058 100644 Binary files a/doc/pub/Regression/pdf/Regression-beamer.pdf and b/doc/pub/Regression/pdf/Regression-beamer.pdf differ diff --git a/doc/pub/Regression/pdf/Regression-minted.pdf b/doc/pub/Regression/pdf/Regression-minted.pdf index 98169661f..7f3c9a457 100644 Binary files a/doc/pub/Regression/pdf/Regression-minted.pdf and b/doc/pub/Regression/pdf/Regression-minted.pdf differ diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt index 77836162d..3ec5e5da7 100644 --- a/doc/src/Regression/Regression.do.txt +++ b/doc/src/Regression/Regression.do.txt @@ -2420,11 +2420,8 @@ we can define \begin{align} \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\ &= \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\ -&= -\begin{cases} -\frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \qquad\qquad\quad \ \ \text{if $h = 0$} \\ -\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \text{else} -\end{cases}. +&= \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} \\ +&=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else} \end{align} !et