Example 12.3 LMS and LTS Univariate (Location) Problem: Barnett and Lewis Data

If you do not specify matrix of the last input argument, the regression problem is reduced to the estimation problem of the location parameter . The following example is described in Rousseeuw and Leroy (1987):

title2 "*** Barnett and Lewis (1978) ***";
b = { 3, 4, 7, 8, 10, 949, 951 };

optn = j(9,1,.);
optn[2]= 3;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */

call lms(sc,coef,wgt,optn,b);

Output 12.3.1 shows the results of the unweighted LS regression.

Output 12.3.2 shows the results for LMS regression.

You obtain the LMS location estimate compared with the mean (which is the LS estimate of the location parameter) and the median . The scale estimate in the univariate problem is a resistant (high breakdown) estimator for the dispersion of the data (see Rousseeuw and Leroy (1987)).

For weighted LS regression, the last two observations are ignored (that is, given zero weights), as shown in Output 12.3.3.

Use the following code to obtain results from LTS:

title2 "*** Barnett and Lewis (1978) ***";
b = { 3, 4, 7, 8, 10, 949, 951 };

optn = j(9,1,.);
optn[2]= 3;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */

call lts(sc,coef,wgt,optn,b);

The results for LTS are similar to those reported for LMS in Rousseeuw and Leroy (1987), as shown in Output 12.3.4.

Since nonzero weights are chosen for the same observations as with LMS, the WLS results based on LTS agree with those based on LMS (shown previously in Output 12.3.3).

In summary, you obtain the following estimates for the location parameter:

• LS estimate (unweighted mean) = 276

• Median = 8

• LMS estimate = 5.5

• LTS estimate = 5.5

• WLS estimate (weighted mean based on LMS or LTS) = 6.4