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\begin{document}

\title{Stats 210A, Fall 2017\\
  Bonus Homework\\
  {\large {\bf Not due}}}
\date{}
\maketitle
\vspace{-5em}


\begin{description}

\item[1. Derived Intervals for $\bar\mu$]\hfill\\
  You are working with a scientist, who has one-way layout data:
\[
y_i = \mu_i + \ep_i, \quad i=1,\ldots,n \quad \text{with}\;\; \ep_i \simiid N(0,1)
\]

The scientist asks you to come up with FWER-controlling confidence intervals for $\mu_1,\ldots,\mu_n$, and you construct
\begin{equation}\label{eq:CI}
C_i = y_i \pm z_{\tilde{\alpha}_n/2}, \quad \text{where} \;\;\tilde{\alpha}_n = 1 - (1 - \alpha)^{1/n} \approx \alpha/n.
\end{equation}

After the scientist sees the results, she notices an interesting fact: even though only a few of the intervals exclude 0, most of the $y_i$ are larger than zero. This makes her curious about the value of the parameter $\bar{\mu} = n^{-1}\sum_{i=1}^n\mu_i$, and she expresses regret that she didn't think of asking about it before seeing the data. 

``Aha!'' you cry, ``but we can use the intervals we just constructed to derive an interval for $\mu$.''
\begin{enumerate}[(a)]
\item Use the simultaneous confidence intervals above to derive a confidence region for $\mu\in\RR^n$.
\item Give an explicit expression akin to \eqref{eq:CI} for the interval for $\bar\mu$ based on this region.
\item What is the approximate asymptotic radius of this interval as $n\to\infty$?
\item What is its radius for $\alpha = 0.05$ and $n =$ 3, 5, 10, and 100? (Give numbers e.g. 3.45).
\end{enumerate}

\item[4. Testing Hypotheses in Fixed Order]\hfill \\
Suppose someone gives us an a priori ordering on hypotheses
$H_{0,1}, \ldots , H_{0,n}$ with $p$-values $p_1, \ldots p_m$ (i.e. the order is specified in advance of looking at the data). 

We then use the following procedure: If $p_1 > \alpha$, stop and accept all null hypotheses. Otherwise, reject $H_{0,1}$ and keep going. Then, if $p_2 > \alpha$, stop and accept $H_{0,2}$ through $H_{0,m}$. Otherwise, reject $H_{0,2}$ and keep going, etc.

In other words, if $k$ is the index of the first p-value that is larger than $\alpha$, we reject $H_{0,i}$ for each $i < k$ and accept the rest.

Prove that this procedure controls the FWER, regardless of the dependence structure of the $p$-values.

\end{description}




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