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?