The asymptotic variance–covariance matrix, Baarda test and the reliability of L1-norm estimates

Abstract

This paper derives the complete Bahadur-type linear representation of the basic vector including the residual vector and the adjusted vector of the observations for the L1-norm estimation. The asymptotic variance–covariance matrix of the basic vector is obtained accordingly. The L1-Baarda test statistic is successfully obtained from the residual and its variance. The reliability, understood as the ability to detect as well as resist outliers, of the L1-norm estimate is discussed for the L1-Baarda test statistic, as compared with the reliability of the L2-norm estimate; it is less affected by the variation of the redundancy component. According to the relationship between residuals and true errors for the L1-norm estimate, the outlier is almost completely projected onto the corresponding residual. This is why the L1-Baarda test seems to locate outliers more correctly. For the L2-norm estimates, the outlier is partly projected onto the corresponding residual at the proportion of the corresponding redundancy component, which may easily result in an error of the third kind - the mis-location of the outlier through the L2-Baarda test. Besides, the robustness of the L1–norm estimate is no longer guaranteed if there exist leverage points in observations. This paper suggests the modified L1-norm estimate for robust estimation of leverage points, and derives its Bahadur-type representation and the corresponding variance–covariance matrix. In numerical examples, the L1-norm estimate and modified L1–norm estimate are compared with the L2–norm estimate, Biber-estimate, Koch’s method and the least median of squares method. The result shows that the L1-norm estimation can better identify small outliers, and that the modified L1-norm estimate is valid for robust estimation in the presence of leverage points.