The following contrived data set (discussed in Chapter 3) is from Anscombe

(1973):

(a) Graph the data and confirm that the third observation is an outlier. Find the leastsquares regression of Y on X, and plot the least-squares line on the graph.

(b) Fit a robust regression to the data using the bisquare or Huber M estimator. Plot the fitted regression line on the graph. Is the robust regression affected by the outlier?

(c) Omitting the third observation f13; 12:74g, the line through the rest of the data has the equation Y ¼ 4 þ 0:345X, and the residual of the third observation from this line is 4.24. (Verify these facts.) Generate equally discrepant observations at X-values of 23 and 33 by substituting these values successively into the equation Y ¼ 4 þ 0:345X þ4:24: Call the resulting Y values Y0 3 and Y00 3 . Redo parts

(a) and (b), replacing the third observation with the point f23; Y0 3g. Then, replace the third observation with the point f33; Y00 3 g. What happens?

(d) Repeat part

(c) using the LTS bounded-influence estimator. Do it again with the MM estimator.