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The code does not work even if I have followed the advices found in this community. What's wrong?

\subsubsection{Score function vs. Influence function}
\begin{multicols}{2}
    \paragraph{Score Function}
    Let $l(\theta|y)$ be the log-likelihood for $\theta \in \R$ and $U_\theta(y) = \dfrac{\partial}{\partial \theta} l(\theta|y)$ be the score function, we have
    \noindent   
    \begin{align*}
        \EE{U_\theta(Y)} &= 0 \\
        \var{\widehat{\theta}} &\simeq \frac{1}{n\var{U_\theta(Y)}} = \\ 
        &= \frac{1}{n \EE{U_\theta^2(Y)} - \EE{U_\theta(Y)}^2} = \\
        &= \frac{1}{n \EE{U_\theta^2(Y)}\\
    \end{align*}
    where $\EE{U_\theta^2(Y)}$ is the \textbf{Fisher Information}.
    
    \columnbreak
    
    \paragraph{Influence Function}
    In nonparametric estimation we have for the influence function that 
    \noindent   
    \begin{align*}
        \EE{L_F(Y)} &= 0 \\
        \var{\widehat{\theta}} &\simeq \frac{\var{L_F(Y)}}{n} = \\
        &= \frac{\EE{L_F^2(Y)} - \EE{L_F(Y)}^2}{n} = \\
        &= \frac{\EE{L_F^2(Y)}}{n}\\
    \end{align*}
\end{multicols}
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    please always provide a small but complete example. We can no run this and you have not said what error you got. Have you loadd the multicol and amsmath packages to define these commands. whae is \EE ? anyone who tries to run this fragment will get multiple errors but we can not guess what error you get. Mar 13, 2022 at 11:00
  • 1
    it will not make an error just bad space (and not affect indentation) but never use \noindent before align Mar 13, 2022 at 11:01

1 Answer 1

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I'm afraid I have no idea how \EE is supposed to be defined. I do have a guess, though, as to how \R and \var might be defined. The following code compiles; however, to make it usable your purposes, you'll need to provide a definition for \EE.

enter image description here

\documentclass{article}
\usepackage{amsmath,amssymb,multicol}
\providecommand\R{\mathbb{R}}
\providecommand\EE[1]{#1} % ???
\DeclareMathOperator{\var}{Var} % variance operator, right?


\begin{document}
\setcounter{section}{5}   % just for this example
\setcounter{subsection}{3}

\subsubsection{Score function vs.\ influence function}

\begin{multicols}{2}

\paragraph{Score Function}
Let $l(\theta\mid y)$ be the log-likelihood for $\theta \in \R$ and $U_\theta(y) = \frac{\partial}{\partial \theta} l(\theta\mid y)$ be the score function. We have  
    \begin{align*}
        \EE{U_\theta(Y)} &= 0 \\
       \var\hat{\theta} &\simeq \frac{1}{n\var{U_\theta(Y)}} = \\ 
       &= \frac{1}{n \EE{U_\theta^2(Y)} - \EE{U_\theta(Y)}^2} = \\
        &= \frac{1}{n \EE{U_\theta^2(Y)}}
    \end{align*}
where $\EE{U_\theta^2(Y)}$ is the \textbf{Fisher Information}.
    
\columnbreak
    
\paragraph{Influence Function}
In nonparametric estimation we have for the influence function that   
    \begin{align*}
        \EE{L_F(Y)} &= 0 \\
        \var\hat{\theta} &\simeq \frac{\var L_F(Y)}{n} = \\
        &= \frac{\EE{L_F^2(Y)} - \EE{L_F(Y)}^2}{n} = \\
        &= \frac{\EE{L_F^2(Y)}}{n}
    \end{align*}

\end{multicols}

\end{document}

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