[SOLVED] Statistics 215b assignment 5

Math stats
Work the following exercises in Efron (2010): 1.1, 1.2, 1.4, 1.5.
Simulation
Produce your own version of Table 1.2 in Efron (2010) by repeating the simulation study described
on pp. 7-9. Use the same i’s as Efron. Explain how many decimal places of agreement one would
expect to see between your results and Efron’s. How well did you meet this expectation?

Shrinking radon
The file srrs2.dat contains 12,777 observed radon levels from households throughout the United
States. This data file comes from Andrew Gelman’s website,
http://www.stat.columbia.edu/~gelman/arm/software/. We will focus on the 766
measurements taken in the basements of the Minnesota homes. These homes are spread across
85 counties in Minnesota; the data set tells us which observations came from which counties.
 Load the data into R. Extract the subset of observations taken in Minnesota basements. Although there is a basement variable, you should instead use the floor variable—a zero
value means a basement. (Don’t ask.)
 Reduce the data set further: keep only the data for counties with at least 10 observations.

You should find 17 such counties, with a total of 511 observations.
 Now split the data into two sets: a training set with five randomly chosen observations from
each county, and a test set with the other observations.
 Compute , the vector of mean radon levels by county in the test data. Radon levels are given
in the variable activity. From now on we will treat  as a population-level parameter to
be estimated.

 Make the standard James-Stein independent-normals assumption: the five observations in
county i are iid draws from a N .i
;  2
/ distribution; these five draws are independent of
the draws from every other county. Compute O
.MLE/
, the maximum-likelihood estimate of 
based on the training data.
 Now compute O
.JS/
, the James-Stein estimator, using the average value in O
.MLE/
as the
shrinkage target. We are assuming that the components of O
.MLE/
share a common SE. Using
the same number of observations in each county tends to aid this assumption. To estimate
this shared SE, you must estimate 
2
, using the pooled-variance technique: add up all the
within-county squared residuals, and divide by the total degrees of freedom.
Caution: The SE of O
.MLE/
i
is not . If you proceed as though it is, you will over-shrink.
 What is the total squared error of O
.MLE/
? Of O
.JS/
? What is the ratio of the larger to the
smaller? What do you conclude about Stein shrinkage in this application?