Question: Consider the following data set, which shows how sensitive the exact fits approach to LMS can be to small perturbations in the data.
Fill in the blank with 9.338 and use the Minitab macro LMSONEX to obtain the LMS solution using the PROGRESS approach. Then change 9.338 to 9.339 and repeat. Notice that the LMS coefficients change dramatically. Now use the Minitab macro EXACTLMS to obtain the exact LMS solution using each of the two numbers. Notice that EXACTLMS produces three "best" solutions for each case, and the solutions differ considerably as well as differing considerably from the PROGRESS solution, although all three solutions for each example produce the same value of the median squared residual. What does this exercise suggest regarding the algorithm that should be used to obtain the LMS solution, and what does it also suggest about using an objective function that is based on the median squared residual?