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\title{The Weibull and Gumbel (Extreme Value) Distributions\footnote{See last slide for copyright information.}}
\subtitle{STA312 Fall 2023}
\date{} % To suppress date


\begin{document}

\begin{frame}
  \titlepage
\end{frame}


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\begin{frame}
\frametitle{The Weibull Distribution}
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
    f(t|\alpha,\lambda) = \left\{ \begin{array}{ll} % ll means left left
   \alpha\lambda(\lambda t)^{\alpha-1} \, \exp\{-(\lambda t)^\alpha \} 
   & \mbox{for $ t \geq 0$}  \\
   0     & \mbox{for } t < 0 
   \end{array}  \right. , % Need that crazy invisible period
\end{displaymath}
} % End size
where $\alpha>0$ and $\lambda>0$.  
\end{frame}

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\begin{frame}
\frametitle{Weibull with $\alpha = 1/2$ and $\lambda=1$}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{w1}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Weibull with $\alpha = 1$ and $\lambda=1$}
\framesubtitle{Standard exponential} 
\begin{center}
\includegraphics[width=2.9in]{w2}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Weibull with $\alpha = 1.5$ and $\lambda=1$}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{w3}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Weibull with $\alpha = 5$ and $\lambda=1$}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{w4}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Weibull with $\alpha = 5$ and $\lambda=1/2$}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{w5}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{The Weibull Distribution}
%\framesubtitle{} 
\begin{displaymath}
    f(t|\alpha,\lambda) = \left\{ \begin{array}{ll} % ll means left left
   \alpha\lambda(\lambda t)^{\alpha-1} \, \exp\{-(\lambda t)^\alpha \} 
   & \mbox{for $ t \geq 0$}  \\
   0     & \mbox{for } t < 0 
   \end{array}  \right. , % Need that crazy invisible period
\end{displaymath}
where $\alpha>0$ and $\lambda>0$.   
\begin{eqnarray*}
    E(T^k) & = & \frac{\Gamma(1+\frac{k}{\alpha})}{\lambda^k} \\  
    \mbox{Median} & = & \frac{[\log(2)]^{1/\alpha}}{\lambda} \\  
    S(t) & = & \exp\{-(\lambda t)^\alpha \} \\  
    h(t) & = & \alpha\lambda^\alpha t^{\alpha-1}
\end{eqnarray*}  \pause

\begin{itemize}
    \item If $\alpha=1$, Weibull reduces to exponential and $h(t)=\lambda$. \pause
    \item If $\alpha>1$, the hazard function is increasing in $t$. \pause
    \item If $\alpha<1$, the hazard function is decreasing.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{The Gumbel (or Extreme Value) Distribution}
\framesubtitle{This version is based on the log of an exponential, not -log as in HW4} 
{\LARGE
\begin{displaymath}
    f(y|\mu,\sigma) = \frac{1}{\sigma} \, \exp\left\{ \left( \frac{y-\mu}{\sigma}\right) 
                                    - e^{ \left(\frac{y-\mu}{\sigma}\right)}  \right\}
\end{displaymath} 
} % End size
where $\sigma>0$. \pause
\begin{itemize}
    \item This is a location-scale family of distributions.  
    \item $\mu$ is the location and $\sigma$ is the scale. \pause
    \item Write $Y \sim G(\mu,\sigma)$.
\end{itemize}
\end{frame}


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\begin{frame}
\frametitle{Log (not $-\log$) of standard exponential is Gumbel(0,1)}
\framesubtitle{$\mu=0$ and $\sigma=1$} 
\begin{center}
\includegraphics[width=2.9in]{Gumbel}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Properties of the $G(0,1)$ Distribution}
\framesubtitle{$f(y) = \exp\left\{ y-e^y  \right\}$ for all real $y$.} \pause
\begin{columns}  
\column{0.35\textwidth}
\includegraphics[width=1.5in]{Gumbel}  
\column{0.65\textwidth}
Let $Z \sim G(0,1)$.  
  \begin{itemize}
    \item MGF is $M_z(t)=\Gamma(t+1)$.  
    \item $E(Z) = \Gamma^\prime(1) = -0.5772157\ldots$   $= -\gamma$, where $\gamma$ is Euler's constant.  
    \item $Var(Z) = \frac{\pi^2}{6}$.  
    \item Median is $\log(\log(2)) = -0.3665129\ldots$  
    \item Mode is zero. 
  \end{itemize}
\end{columns}
\end{frame}


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\begin{frame}
\frametitle{General  $Y \sim G(\mu,\sigma)$}
\framesubtitle{$f(y|\mu,\sigma) = \frac{1}{\sigma} \, \exp\left\{ \left( \frac{y-\mu}{\sigma}\right) - e^{ \left(\frac{y-\mu}{\sigma}\right)}  \right\}$}  \pause
Let $Z \sim G(0,1)$ and $Y = \sigma Z + \mu$.   Then $Y \sim G(\mu,\sigma)$. \pause
\begin{itemize}
    \item $E(Y) = \sigma E(Z) + \mu = \mu - \sigma\gamma$.  
    \item $Var(Y) = \sigma^2 Var(Z) = \sigma^2 \frac{\pi^2}{6}$.  
    \item Median is $\sigma\log\log(2) + \mu$.  
    \item Mode is $\mu$. 
\end{itemize}
\end{frame}


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\begin{frame}
\frametitle{Log (not minus log) of Weibull is Gumbel} 
%\framesubtitle{} 
\begin{itemize}
    \item Let $T \sim$ Weibull($\alpha,\lambda$), and $Y=\log(T)$. \pause
    \item In addition, re-parameterize, meaning express the parameters in a different, equivalent way.  \pause
    \item Let  $\sigma = \frac{1}{\alpha}$ and $\mu=-\log\lambda$. 
    \item Or equivalently, substitute $\frac{1}{\sigma}$ for $\alpha$ and $e^{-\mu}$ for $\lambda$. \pause
    \item The result is $Y \sim G(\mu,\sigma)$. \pause
    \item[]
    \item So if you believe the distribution of a set of failure time data could be Weibull (a popular choice), you can log-transform the data and apply a Gumbel model. \pause
    \item The Gumbel distribution may be preferable because the parameters $\mu$ and $\sigma$ are easy to interpret.
\end{itemize}
\end{frame}


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\begin{frame}
\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistics, 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:
\href{http://www.utstat.toronto.edu/brunner/oldclass/312f23} {\footnotesize \texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f23}}

\end{frame}


\end{document}



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\begin{frame}
\frametitle{}\pause
%\framesubtitle{} 
  \begin{itemize}
    \item \pause
    \item \pause
    \item 
  \end{itemize}
\end{frame}

{\LARGE
\begin{displaymath}
    \log \widehat{S}(t) = \sum_{t_j \leq t} \log \widehat{p}_j
\end{displaymath}  \pause
} % End size


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# Making the pictures of Weibull
t = seq(from=0,to=2.5,length=101)
alpha = 5; lambda = 1/2
Density = dweibull(t,shape=alpha,scale=1/lambda)
plot(t,Density,type='l',ylim=c(0,2.5))
tstring = paste('Weibull Density with alpha =',alpha,' and lambda =',lambda)
title(tstring)

# Gumbel plot
y = seq(from=-5,to=4,length=101)
Density = exp(y-exp(y))
plot(y,Density,type='l')
title('Standard Gumbel Density')