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\title{Stat 471: Homework 06 }
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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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{\bf Due Date:} December 8, 1:20pm. No late assignments will be
accepted or graded. There will be $10\%$ bonus points for
exceptional solutions.


\begin{enumerate}

\item Using $\tt{strength.data}$, fit model
$$\tt{grip} = g(\tt{arm}) + \epsilon$$
using Gaussian kernel smoothing. Choose your bandwidth $\sigma^2$ by mimimizing leave--one-out cross-validation criterion. 

\item  (Problem due to M. Newton) There is an investigator at NIH who may be interested.to compute the probability distribution of

$$ R_{2k} =  \frac{X_{(2k)}}{X_{(k)}}        $$

where $X_{(k)} = \max (X_1,...,X_k)$ is an order statistic of $X_i \sim \mbox{Gamma}(\alpha,\beta)$
He is curious to know how sensitive the distribution is to the baseline distribution of the $X_i$'s. Make a $\tt{MATLAB}$ function that takes $\alpha,\beta,k$ as input variables and generate the kernel density estimate $\tt{fhat}$ (see lecture 31 $\tt{MATLAB}$ output). Plot $\hat f$ for various $\alpha$ and $\beta$. How sensitivy $\hat f$ is when you change $\alpha$ and $\beta$? You may assume $k$ to be very large, i,e. $k=100$ and use the built-in Gamma random number generator.

\end{enumerate}
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