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\title{Estimation - Introduction}
\author{Aubin SIONVILLE}
\date{Télécom St Etienne 2024-2025}

\maketitle

\center

\section*{Définitions}

\subsection*{$N$-échantillon aléatoire}
\[\boxed{\begin{array}{l}
    N\text{-échantillon aléatoire issu d'une v.a. } X :\\
    \text{Ensemble } \{X_1, ..., X_N\} \text{ de v.a. i.i.d.}
\end{array}}\]

\vspace*{0.3cm}

\begin{multicols}{2}
    \subsection*{Estimation}
    \[\boxed{\begin{array}{l}
        \text{Estimation } \widehat\theta \text{ de la "vraie valeur" } \theta^* :\\
        \displaystyle \widehat\theta = f(x_1, ..., x_N) \quad \textit{Exemple : } \widehat\theta = \frac{1}{N} \sum_{n=1}^{N}x_n
    \end{array}}\]

    \subsection*{Estimateur}
    \[\boxed{\begin{array}{l}
        \text{Estimateur associé : la v.a. }\\
        \displaystyle \widehat\Theta = f(X_1, ..., X_N) \quad \textit{Exemple : } \widehat\Theta = \frac{1}{N} \sum_{n=1}^{N}X_n
    \end{array}}\]
\end{multicols}

\vspace*{0.5cm}

\section*{Exemple de grandeurs statistiques}

\textit{Formules à (peut-être) ne pas apprendre par cœur}

\textit{Plusieurs estimateurs possibles : lequel choisir ?}

\begin{multicols}{3}
    \subsection*{Moyenne}
    \[\boxed{\displaystyle \widehat\mu = \frac{1}{N}\sum_{n=0}^{N-1}x[n]}\]

    \subsection*{Moments centraux}
    \[\boxed{\displaystyle \widehat\mu^c_k = \frac{1}{N}\sum_{n=0}^{N-1}(x_n - \widehat\mu)^k}\]

    \subsection*{Variance}
    \[\boxed{\displaystyle \widehat\sigma^2 = \frac{1}{N-1}\sum_{n=0}^{N-1}(x_n - \widehat\mu)^2}\]    
\end{multicols}

\begin{multicols}{2}
    \subsection*{Skewness}
    \[\boxed{\displaystyle \widehat\gamma_1 = \Bigg(\frac{\sqrt{N(N-1)}}{N-2}\Bigg)\frac{\widehat\mu^c_3}{\big(\widehat\mu^c_2\big)^{3/2}}}\]
    
    \subsection*{Kurtosis}
    \[\boxed{\displaystyle \widehat\gamma_2 = \bigg(\frac{(N-1)(N+1)}{(N-2)(N-3)}\bigg)\frac{\widehat\mu^c_4}{\big(\widehat\mu^c_2\big)^2} - 3\bigg(\frac{(N-1)^2}{(N-2)(N-3)}\bigg)}\]
\end{multicols}

\begin{multicols}{2}
    \subsection*{Autocorrélation}
    \[\boxed{\begin{array}{lll}\displaystyle \widehat{r}_{xx}[k] & = & \displaystyle \frac{1}{N}\sum_{n=0}^{N-1}x[n]x[n+k] \\
       OU & = & \displaystyle \frac{1}{N - |k|}\sum_{n=0}^{N-1}x[n]x[n+k]
    \end{array}}\]

    \subsection*{Densité spectrale de puissance}
    \[\boxed{\begin{array}{lll}\displaystyle \mathrm{Per}[\nu_k] & = & \displaystyle \frac{1}{N}|\mathrm{TFD}[\nu_k]|^2\\
        & = & \displaystyle \frac{1}{N} \Bigg|\sum_{n=0}^{N-1}x[n]e^{-j2\pi\nu_kn}\Bigg|^2
    \end{array}}\]

\end{multicols}

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