Tslatex Rayanne Lenox 〈2024-2026〉

\sectionMaximum Likelihood Estimation

\subsectionPart (a): Derive the log-likelihood Given $y_i \sim \mathcalN(\mu, \sigma^2)$ i.i.d., the log-likelihood is:

\begindocument \maketitle

\usepackagerayanne_macros

% Matrices \beginpmatrix a & b \ c & d \endpmatrix

\beginalign \ell(\mu, \sigma^2) &= \sum_i=1^n \log f(y_i \mid \mu, \sigma^2) \ &= -\fracn2\log(2\pi) - \fracn2\log\sigma^2 - \frac12\sigma^2\sum_i=1^n (y_i - \mu)^2 \labeleq:loglik \endalign

\ProvidesPackagerayanne_macros \RequirePackagetslatex \newcommand\indep\perp!!!\perp \newcommand\E\mathbbE \renewcommand\P\mathbbP \DeclareMathOperator\VarVar \DeclareMathOperator\CovCov TsLatex Rayanne Lenox

\sectionQuestion 1 Your solution here.

% Text in math \textsubject to % inside \text{}

\begindocument \maketitle

% Cases f(x) = \begincases 0 & x<0 \ 1 & x\ge 0 \endcases Create rayanne_macros.sty :

Then in your main file:

\subsectionPart (b): First-order conditions Taking the derivative w.r.t. $\mu$: TsLatex Rayanne Lenox

\beginalign \frac\partial \ell\partial \mu = \frac1\sigma^2\sum_i=1^n (y_i - \mu) \stackrelset= 0 \ \implies \hat\mu_MLE = \bary \endalign