Algorithm

The initial LTS estimate is computed using the algorithm described in the section LTS Estimate. You can control the quantile of the LTS estimate with the option INITH=, where is an integer between and . By default, , which corresponds to a breakdown value of around .

The initial S estimate is computed using the algorithm described in the section S Estimate. You can control the breakdown value and efficiency of this initial S estimate by the constant , which can be specified with the K0 option.

The scale parameter is solved by an iterative algorithm

where .

Once the scale parameter is computed, the iteratively reweighted least squares (IRLS) algorithm with fixed scale parameter is used to compute the final MM estimate.

Convergence Criteria

In the iterative algorithm for the scale parameter, the relative change of the scale parameter controls the convergence.

In the iteratively reweighted least squares algorithm, the same convergence criteria for the M estimate used before are used here.

Bias Test

Although the final MM estimate inherits the high breakdown value property, its bias due to the distortion of the outliers can be high. Yohai, Stahel, and Zamar (1991) introduced a bias test. The ROBUSTREG procedure implements this test when you specify the BIASTEST option in the PROC statement. This test is based on the initial scale estimate and the final scale estimate , which is the solution of

Let and . Compute

Let

Standard asymptotic theory shows that approximately follows a distribution with degrees of freedom. If exceeds the quantile of the distribution with degrees of freedom, then the ROBUSTREG procedure gives a warning and recommends that you use other methods. Otherwise the final MM estimate and the initial scale estimate are reported. You can specify with the ALPHA= option following the BIASTEST option. By default, ALPHA=0.99.

Asymptotic Covariance and Confidence Intervals

Since the MM estimate is computed as a M estimate with a fixed scale in the last step, the asymptotic covariance for the M estimate can be used here for the asymptotic covariance of the MM estimate. Besides the three estimators H1, H2, and H3 as described in the section Asymptotic Covariance and Confidence Intervals, a weighted covariance estimator H4 is available:

where is the correction factor and , .

You can specify these estimators with the option ASYMPCOV= [H1 | H2 | H3 | H4]. The ROBUSTREG procedure uses H4 as the default. Confidence intervals for estimated parameters are computed from the diagonal elements of the estimated asymptotic covariance matrix.

R Square and Deviance

The robust version of for the MM estimate is defined as

and the robust deviance is defined as the optimal value of the objective function on the scale:

where , is the MM estimator of , is the MM estimator of location, and is the MM estimator of the scale parameter in the full model.

Linear Tests

For MM estimation, the same test and test used for M estimation can be used. See the section Linear Tests for details.

Model Selection

For MM estimation, the same two model selection methods used for M estimation can be used. See the section Model Selection for details.