% 431s23Formulas6.tex
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\begin{center}   
{\Large \textbf{STA 431 Formulas}}\\  
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\noindent
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\begin{tabular}{ll}
$ Pr\{ Y \in A\} = \sum_x Pr\{ Y \in A|X=x\} p_x(x)$ &
$ Pr\{ Y \in A\} = \int_{-\infty}^\infty Pr\{ X \in A|X=x\} f_x(x) \, dx$ \\
$Var(X) \stackrel{def}{=} E\left( \, (X-\mu_x)^2 \, \right) $ &
$Var(X) = E(X^2)-[E(X)]^2 $  \\ 
$Cov(X,Y) \stackrel{def}{=} E\left( \, (X-\mu_x)(Y-\mu_{_Y}) \, \right)$    &
$Cov(X,Y) = E(XY)-E(X)E(Y)$ \\
$Cov(aX,bY) = ab \, Cov(X,Y)$ & $Cov(x+a,Y+b) = Cov(X,Y)$ \\
$Corr(X,Y) \stackrel{def}{=} \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)} } $ & 
$Cov\left(\sum_{i=1}^na_iX_i~,\,\sum_{j=1}^m b_j Y_j \right) = 
\sum_{i=1}^n\sum_{j=1}^m  a_i b_j Cov\left( X_i, Y_j \right)$ \\
%%%%%%%%%%%
$\mathbf{Av} = \lambda\mathbf{v}$ &
$\mathbf{A} = \mathbf{CDC}^\top$ with $\mathbf{CC}^\top = \mathbf{C}^\top\mathbf{C} = \mathbf{I}$, 
 for $\mathbf{A}$ symmetric.\\
$\mathbf{A}^{-1} = \mathbf{C} \mathbf{D}^{-1} \mathbf{C}^\top$ & \\
$\mathbf{A}^{1/2} = \mathbf{CD}^{1/2} \mathbf{C}^\top$ & 
$\mathbf{A}^{-1/2} = \mathbf{CD}^{-1/2} \mathbf{C}^\top$  \\
\multicolumn{2}{l}{$\mathbf{A}$ \emph{positive definite} means  $\mathbf{v}^\prime \mathbf{Av} > 0$ for all vectors $\mathbf{v} \neq \mathbf{0}$.} \\
%%%%%%%%%%%
$cov(\mathbf{x}) \stackrel{def}{=} 
E\left\{(\mathbf{x}-\boldsymbol{\mu}_x)(\mathbf{x}-\boldsymbol{\mu}_x)^\top\right\}$ &
$cov(\mathbf{x,y}) \stackrel{def}{=} E\left\{ (\mathbf{x}-\boldsymbol{\mu}_x)
                             (\mathbf{y}-\boldsymbol{\mu}_y)^\top\right\}$ \\
$cov(\mathbf{Ax}) = \mathbf{A} \, cov(\mathbf{x}) \, \mathbf{A}^\top$ &
$cov(\mathbf{Ax},\mathbf{By}) = \mathbf{A}  \, cov(\mathbf{x,y}) \,  \mathbf{B}^\top$
\\
$\mathbf{L} = \mathbf{A}_1\mathbf{X}_1 + \cdots +  \mathbf{A}_m\mathbf{X}_m  + \mathbf{b}$ &
$cov(\mathbf{L}_1,\mathbf{L}_2) =  
\sum_{i=1}^m \sum_{j=1}^n \mathbf{A}_i \, cov(\mathbf{x}_i,\mathbf{y}_j) \, \mathbf{B}_j^\top$  
\\
\multicolumn{2}{l} {If $\mathbf{x} \sim N_p(\boldsymbol{\mu},\boldsymbol{\Sigma} )$, then 
$\mathbf{Ax} + \mathbf{b} \sim N_r(\mathbf{A}\boldsymbol{\mu} + \mathbf{b},
                    \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^\top )$ and 
$(\mathbf{x}-\boldsymbol{\mu})^\top
           \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu}) \sim \chi^2(p)$} \\
\multicolumn{2}{l} {$L(\boldsymbol{\mu,\Sigma}) =     |\boldsymbol{\Sigma}|^{-n/2} (2\pi)^{-np/2} 
    \exp -\frac{n}{2}\left\{ tr(\boldsymbol{\widehat{\Sigma}\Sigma}^{-1}) +
    (\overline{\mathbf{x}}-\boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} 
    (\overline{\mathbf{x}}-\boldsymbol{\mu}) \right\}$   } \\
    $\boldsymbol{\widehat{\mu}} =  \overline{\mathbf{x}}$, ~~
    $\boldsymbol{\widehat{\Sigma}} = 
    \frac{1}{n}\sum_{i=1}^n (\mathbf{x}_i-\overline{\mathbf{x}}) 
                        (\mathbf{x}_i-\overline{\mathbf{x}})^\top $ & 
    $\widehat{\sigma}^2_x = \frac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2$, ~~
    $\widehat{\sigma}_{xy} = \frac{1}{n} \sum_{i=1}^n (x_i-\overline{x})(y_i-\overline{y})$ \\
%%%%%%%%%%%
\multicolumn{2}{l} { If $\mathbf{t}_n \stackrel{p}{\rightarrow} \mathbf{c}$ and $g(\mathbf{x})$ is continuous at $\mathbf{x}=\mathbf{c}$, then $g(\mathbf{t}_n) \stackrel{p}{\rightarrow} g(\mathbf{c})$. } \\
\multicolumn{2}{l} { If $\mathbf{x}_1, \ldots, \mathbf{x}_n \stackrel{iid}{\sim} (\boldsymbol{\mu},\boldsymbol{\Sigma})$, then $\overline{\mathbf{x}}_n \stackrel{p}{\rightarrow} \boldsymbol{\mu}$ and $\overline{\mathbf{x}}_n \stackrel{\cdot}{\sim} N_p(\boldsymbol{\mu}, \frac{1}{n}\boldsymbol{\Sigma})$ } \\  
\multicolumn{2}{l} {$\widehat{\boldsymbol{\theta}}_n \stackrel{p}{\rightarrow} \boldsymbol{\theta}$ and $\widehat{\boldsymbol{\theta}}_n \stackrel{\cdot}{\sim} N_m(\boldsymbol{\theta},\mathbf{V}_n)$, with 
$\widehat{\mathbf{V}}_n = \mathbf{H}^{-1}$, where $\mathbf{H} = \left[\frac{\partial^2 (-\ell)} {\partial\theta_i\partial\theta_j}\right]_{\boldsymbol{\theta}=\widehat{\boldsymbol{\theta}}}$ } \\
\multicolumn{2}{l} {
 $G^2 = -2 \ln \left( \frac{\max_{\theta \in \Theta_0} L(\theta)}
                           {\max_{\theta \in \Theta}   L(\theta)  }  \right) 
     = -2 \ln \left( \frac{L(\widehat{\theta}_0)}{  L(\widehat{\theta})  }\right)
       \sim \chi^2(r)$ if $H_0: \theta \in \Theta_0$ is true.}   \\
\multicolumn{2}{l} {$W_n = (\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})^\top 
\left(\mathbf{L} \widehat{\mathbf{V}}_n \mathbf{L}^\top \right)^{-1} 
(\mathbf{L}\widehat{\boldsymbol{\theta}}_n-\mathbf{h})
\stackrel{\cdot}{\sim} \chi^2(r)$ 
if $H_0: \mathbf{L}\boldsymbol{\theta} = \mathbf{h}$ is true.} \\
\multicolumn{2}{l} {The reliability of an observable variable $d$ as a measurement of a latent variable $F$ is $\rho^2 \stackrel{def}{=} \left(Corr(F,d)\right)^2$.
} \\
\multicolumn{2}{l} {If $d = F + e$ with $Var(F)=\phi$ and $Var(e)=\omega$, then  
$\rho^2 = \frac{\phi}{\phi+\omega}$.
} \\
\multicolumn{2}{l} {For symmetric $\boldsymbol{\Sigma}_{k \times k}$, there are $k(k+1)/2$ unique elements and  $k(k-1)/2$ unique off-diagonal elements.}
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\noindent
\begin{tabular}{ll}
\multicolumn{2}{c} {\sffamily The Double Measurement Model in centered form:~~~~~~~~~~~~} \\
$\mathbf{y}_i = 
\boldsymbol{\beta} \mathbf{x}_i +  \boldsymbol{\epsilon}_i$ &
$cov(\mathbf{x}_i)=\boldsymbol{\Phi}_x$, $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$ \\
${\mathbf{F}}_i = \left( \begin{array}{c}
{\mathbf{x}}_i  \\ \hline  {\mathbf{y}}_i \end{array} \right)$ &
\hspace{-2.4mm}\begin{tabular}{l}
$\mathbf{x}_i$ is $p \times 1$, $\mathbf{y}_i$ is $q \times 1$, $\mathbf{F}_i$ 
is $(p+q) \times 1$ \\
$cov(\mathbf{F}_i) = \boldsymbol{\Phi}$
\end{tabular} \\
${\mathbf{d}}_{i,1} = {\mathbf{F}}_i + \mathbf{e}_{i,1}$ &
$cov(\mathbf{e}_{i,1})=\boldsymbol{\Omega}_1$, 
$cov(\mathbf{e}_{i,2})=\boldsymbol{\Omega}_2$ \\
${\mathbf{d}}_{i,2} = {\mathbf{F}}_i + \mathbf{e}_{i,2}$ &
$\mathbf{x}_i$, $\boldsymbol{\epsilon}_i$, $\mathbf{e}_{i,1}$ and 
$\mathbf{e}_{i,2}$ are independent. \\
\multicolumn{2}{c} {\sffamily The General Structural Equation Model in centered form: ~~~~~~} \\
$\mathbf{y}_i = \boldsymbol{\beta} \mathbf{y}_i + \boldsymbol{\Gamma} \mathbf{x}_i + \boldsymbol{\epsilon}_i$ &
$cov(\mathbf{x}_i)=\boldsymbol{\Phi}_x$ and $cov(\boldsymbol{\epsilon}_i)=\boldsymbol{\Psi}$\\
$\mathbf{F}_i = \left( \begin{array}{c}
         \mathbf{x}_i  \\ \hline \mathbf{y}_i 
             \end{array} \right)$ &
$cov(\mathbf{F}_i) = \boldsymbol{\Phi}
= \left( \begin{array}{c | c}
         \boldsymbol{\Phi}_{11}  & \boldsymbol{\Phi}_{12} \\ \hline
         \boldsymbol{\Phi}_{12}^\top  & \boldsymbol{\Phi}_{22} \\
             \end{array} \right)$ \\
$\mathbf{d}_i = \boldsymbol{\Lambda}\mathbf{F}_i + \mathbf{e}_i$ &
$cov(\mathbf{e}_i) = \boldsymbol{\Omega}$ \\
$\mathbf{x}_i$, $\boldsymbol{\epsilon}_i$ and $\mathbf{e}_i$ are independent. &
$\mathbf{x}_i$ is $p \times 1$, $\mathbf{y}_i$ is $q \times 1$, $\mathbf{d}_i$ is $k \times 1$. \\
$\boldsymbol{\Phi}_x$ and $\boldsymbol{\Psi}$ are positive definite. & 
% Not $\boldsymbol{\Omega}$ because of the trick of adding observed variables. \\
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% 
\begin{verbatim}

> df = 1:8; CriticalValue = qchisq(0.95,df)
> round(rbind(df,CriticalValue),3)

               [,1]  [,2]  [,3]  [,4]  [,5]   [,6]   [,7]   [,8]
df            1.000 2.000 3.000 4.000  5.00  6.000  7.000  8.000
CriticalValue 3.841 5.991 7.815 9.488 11.07 12.592 14.067 15.507
\end{verbatim}

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This formula sheet was prepared by  \href{http://www.utstat.toronto.edu/brunner}{Jerry Brunner},
Department of Statistics, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:

\begin{center}
\href{http://www.utstat.toronto.edu/brunner/oldclass/431s23} {\texttt{http://www.utstat.toronto.edu/brunner/oldclass/431s23}}
\end{center}


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