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% Survival Analysis Notes ************************************************
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% File Name:      SurvivalAnalysisNotes.tex
% Submitted:      Dr. Eric Suess,Statistics 6502, Spring 2002
% Submitted Date: February 1st, 2002
% Accepted Date:  February 1st, 2002
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\pagestyle{headings} \markright{Statistics 6502, Spring 2002}

\large
\begin{center}
{\bf Nonparametric Bootstrap}
\end{center}

{\bf Empirical cumulative distribution function (ecdf)}

{\bf Def:} For a batch of numbers $x_1, x_2, ... , x_n$ we define
the ecdf as

\begin{equation}
F_n(x) = \frac{1}{n}(\# x \leq x).
\end{equation}

{\bf Ex.} Suppose we have the following batch of numbers: 1, 14,
10, 9, 11, 9.  Draw the ecdf for this batch of numbers.

{\bf Introduction to the Nonparametric Bootstrap}

The Nonparametric Bootstrap is useful for estimating the
variability of a location parameter.

Note: nonparametric means distribution-free.

Suppose $x_1, x_2, ... , x_n$ are observations from an unknown
distribution $F$, it is appropriate to investigate the variability
and sampling distribution of a location estimate calculated from
the sample.

Let $\hat{\theta} = \hat{\theta}(x_1, x_2, ... , x_n)$ be a
location estimate.

{\bf Problems:}
\begin{enumerate}
\item We do not know $F$ and don't necessarily want to make a
parametric assumption about $F$.
\item $\hat{\theta}$ may be complicated. (median, trimmed mean,
etc.)
\end{enumerate}

The nonparametric bootstrap proceeds as follows:

\begin{enumerate}
\item Use the ecdf $F_n(x)$ as an approximation to $F(x)$.
\item Take $B$ samples of size $n$ from $F_n(x)$, {\it with
replacement}.
\begin{enumerate}
\item Estimate the sampling distribution of $\hat{\theta}$ by
plotting a histogram of
\begin{equation}
\hat{\theta}^*_1, \hat{\theta}^*_2, ... ,\hat{\theta}^*_B
\end{equation}
\item Estimate the variability by calculating the sd
\begin{equation}
s_{\hat{\theta}} = \sqrt{\frac{\sum_{i=1}^B ( \hat{\theta}^*_i -
\bar{\hat{\theta}}^* )^2}{B-1}}
\end{equation}
\end{enumerate}
\end{enumerate}

{\bf Example:} Heat of Sublimation of Platinum.  Find the sampling
distribution of the mean, median, and 10\% trimmed mean.

\textit{Reference}:

\hspace{0.25in} Rice, \textit{Mathematical Statistics, 2ed},
Duxbury, 1995.

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