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{\Large \textbf{Sample Questions: Log-Normal Regression}}%\footnote{}
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STA312 Fall 2023. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item Let the continuous random variable $T$ have median $m$. Let $Y = g(T)$, where $g(x)$ is an increasing function. Show that the median of $Y$ is $g(m)$. This is why the median of a log-normal is $e^\mu$. \vspace{100mm}

\item Show that the expected value of a log-normal is $e^{\mu+\frac{1}{2}\sigma^2}$. Hint: the moment-generating function of a normal random variable is $e^{\mu t +\frac{1}{2}\sigma^2 t^2}$.

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\item Write the log-normal regression model in multiplicative form. \vspace{50mm}

\item For a log-normal regression model, show that if $x_{i,k}$ is increased by $c$ units, $E(t_i)$ is multiplied by {\Large $e^{c\beta_k}$}. \vspace{100mm}

\item If $x_{i,k}$ is increased by one unit, the median of $t_i$ is multiplied by \underline{\hspace{15mm}}. \vspace{20mm}

\item If $x_{i,k}$ is increased by one unit, the \emph{value} of $t_i$ is multiplied by \underline{\hspace{15mm}}.

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\item Write the hazard function of a log-normal regression model in terms of $\Phi(x)$, the cumulative distribution function of a standard normal. Is this a proportional hazards model? \vspace{100mm}

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\item Show that in general, if 
$\widehat{\boldsymbol{\theta}}_n \stackrel{.}{\sim} N_k(\boldsymbol{\theta},\mathbf{V}_n)$ and $\mathbf{a}$ is a non-zero $k \times 1$ vector of constants, then 
$W = \mathbf{a}^\top \widehat{\boldsymbol{\theta}}_n \stackrel{.}{\sim}
N\left(\mathbf{a}^\top \boldsymbol{\theta}, \, 
\mathbf{a}^\top \mathbf{V}_n \, \mathbf{a} \right)$. 

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\item What is the parameter vector $\boldsymbol{\theta}$ for a log-normal regression model with $p-1$ explanatory variables? \vspace{20mm}

\item For a log-normal regression model, let $\mathbf{x}_{n+1}$ be a $p \times 1$ vector of explanatory variable values, maybe starting with a 1 for the intercept. A new observation (log failure time) could be written 
$y_{n+1} = \mathbf{x}^\top \boldsymbol{\beta} + \epsilon_{n+1}$, where $\epsilon_{n+1} \sim N(0,\sigma^2)$, and $\epsilon_{n+1}$ is independent of $\epsilon1, \ldots, \epsilon_n$. It is natural to predict the value of $y_{n+1}$ with the estimated expected value, so $\widehat{y}_{n+1} = \mathbf{x}^\top \widehat{\boldsymbol{\beta}}$.

Let $\mathbf{V}_n$ denote the $(p+1) \times (p+1)$ asymptotic covariance matrix of the parameter vector. What is the asymptotic distribution of $\widehat{y}_{n+1}$? 

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\item What is the asymptotic distribution of the error in prediction $y_{n+1} - \widehat{y}_{n+1}$? Justify your answer; include calculation of the expected value and variance.  \vspace{130mm}


\item What is the standard error of $y_{n+1} - \widehat{y}_{n+1}$. Remember, a standard error is an \emph{estimated} standard deviation, something that can be computed from sample data.

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\item Dividing $y_{n+1} - \widehat{y}_{n+1}$ by its standard error, obtain a $Z$ statistic. What is the asymptotic distribution of $Z$? \vspace{30mm}

\item Use the $Z$ statistic to obtain a 95\% prediction interval for $y_{n+1}$. 

\end{enumerate} % End of all the questions

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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/312f23} {\small\texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f23}}
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