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,...

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 = 4 + 0:345 X, 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 = 4 + 0:345 X + 4:24: Call the resulting Y values Y’
₃
and Y’’
₃. Redo parts (a) and (b), replacing the third observation with the point {23; Y’
₃}. Then, replace the third observation with the point f33; Y’’
₃
g. What happens?

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