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{\Large \textbf{Sample Questions: Maximum Likelihood Part 2}}%\footnote{}
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STA312 Spring 2019. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item Let $X_1, \ldots, X_n$ be independent $N(\mu,\sigma^2)$ random variables. 
    \begin{enumerate} 

        \item Derive formulas for the maximum likelihood estimates of $\mu$ and $\sigma^2$. We will establish that it's a maximum later. Show your work and \textbf{circle your final answer}. 

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        \item Calculate the Hessian of the minus log likelihood function: $\mathbf{H} = \left[\frac{\partial^2 (-\ell)} {\partial\theta_i\partial\theta_j}\right]$. Show your work. 


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        \item Give $\widehat{\mathbf{V}}_n$, the estimated asymptotic variance-covariance matrix of the MLE. Show some work. 

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        \item Consider a large-sample $Z$-test of $H_0:\mu=\mu_0$. Give an explicit formula for the test statistic. This is something you would be able to compute with a calculator given $\widehat{\mu}$ and $\widehat{\sigma}^2$. \vspace{100mm}

        \item Consider a large-sample $Z$-test of $H_0:\sigma^2=\sigma^2_0$. Give an explicit formula for the test statistic. This is something you would be able to compute with a calculator. 

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        \item Consider the large-sample likelihood ratio test of $H_0: \mu=\mu_0$.  Derive an explicit formula for the test statistic $G^2$. Show your work and \emph{keep simplifying!}.

    \end{enumerate} % End of the first st of normal questions

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% I had lifted the dead pixels problem: Q7 of 2101f18 A5. But it took us too far afield. Stay with normal.

\item The file \href{http://www.utstat.toronto.edu/~brunner/data/legal/normal.data.txt}
                    {\texttt{http://www.utstat.toronto.edu/$\sim$brunner/data/legal/normal.data.txt}}
has a random sample from a normal distribution. 

    \begin{enumerate}
        \item Find the maximum likelihood estimates of $\widehat{\mu}$ and $\widehat{\sigma}^2$ numerically. Compare the answer to your closed-form solution.
        \item Show that the minus log likelihood is indeed minimized at $(\widehat{\mu}, \widehat{\sigma}^2)$ for this data set.
        \item Calculate the estimated asymptotic covariance matrix of the MLEs.
        \item Give a ``better" estimated asymptotic covariance matrix based on your closed-form solution.
        \item Calculate a large-sample 95\% confidence interval for $\sigma^2$.
        \item Test $H_0: \mu = 103$ with a
            \begin{enumerate}
                \item $Z$-test.
                \item Likelihood ratio chi-squared test. Compare the closed-form version.
                \item Wald chi-squared test.
            \end{enumerate}
             Give the test statistic and the $p$-value for each test.
        \item The coefficient of variation (used in sample surveys and business statistics) is the standard deviation divided by the mean.  
            \begin{enumerate}
                \item Show that multiplication by a positive constant does not affect the coefficient of variation. This is a paper and pencil calculation.
                \item Give a numerical point estimate of the coefficient of variation for  the normal data of this question. Actually, it's the maximum likelihood estimate, because \emph{the invariance principle of maximum likelihood estimation says that the MLE of a function is that function of the MLE}.
                \item Using the delta method, give a 95\% confidence interval for the coefficient of variation. Start with a paper and pencil calculation of \.{g}$(\boldsymbol{\theta}) = \left( \frac{\partial g}{\partial\theta_1}, \ldots ,  \frac{\partial g}{\partial\theta_k} \right)$.
            \end{enumerate}
    \end{enumerate}






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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, 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: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/312s19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/312s19}}
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        \item Do you reject the null hypothesis? Answer Yes or No.
        \item Are the results statistically significant? Answer Yes or No.
        \item Do these data contradict  claim that $\theta = 1.16$? Answer Yes or No.