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Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
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\begin{center}   
{\Large \textbf{STA 431 Quiz 4}}\\
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\noindent
% \emph{Calculators are allowed.} 

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\begin{enumerate} 

\item (5 points) \label{ztest} Suppose you estimate the $m \times 1$ parameter vector $\boldsymbol{\theta}$ by numerical maximum likelihood, obtaining the vector of MLEs $\widehat{\boldsymbol{\theta}}_n$ and the estimated asymptotic covariance matrix $\widehat{\mathbf{V}}_n$. Give a $z$ statistic for testing $H_0: \mathbf{a}^\top \boldsymbol{\theta} = 0$, where $\mathbf{a}$ is an $m \times 1$ non-zero vector of constants. Start with the asymptotic distribution of $\mathbf{a}^\top \widehat{\boldsymbol{\theta}}_n$. Your final answer is a formula for $z$. \textbf{Circle the formula}.

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\item (2 points) Starting with your answer to Question \ref{ztest}, show that for  $H_0: \mathbf{a}^\top \boldsymbol{\theta} = h$, the Wald test statistic is $W_n = z^2$.

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\item (3 points) \label{R} In Question 3 of this week's assignment, you calculated a 95\% confidence interval for the quantity $2\alpha-\beta$, where $\alpha$ and $\beta$ were the parameters of a beta distribution. Your answer was a set of two numbers, the lower confidence limit and the upper confidence limit. \emph{Write the numbers in the space below.}

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On your printout, circle the numbers and write ``Question~\ref{R}" beside them. \emph{Do not answer this question if you do not have a printout.}



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\noindent
\textbf{Please attach your printout to the quiz paper. The printout should show your \emph{complete} R input and output.} Make sure your name and student number appear on the printout.

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