Showing papers by "Joseph W. McKean published in 1997"

7 Rank-based analyses of linear models

Abstract: Publisher Summary This chapter presents the rank-based analyses of linear models. These methods are based on robust estimates of regression parameters in the same way as the traditional analysis of variance (ANOVA) is based on least squares (LS) estimates. It also discusses two classes of robust estimates: regular R-estimates and generalized rank (GR)-estimates. The first class contains highly efficient estimates but their influence is only bounded in the Y space. The second class, although not as efficient, has bounded influence in both the Y and the X spaces. The chapter also presents the rank-based analysis for regular R-estimates and GR-estimates, respectively. The rank-based analysis offers the tests of general linear hypotheses and related inference procedures for all the models covered by the traditional ANOVA and analysis of covariance methods based on LS-estimates. These rank-based analyses are generalizations of the nonparametric procedures in the simple location problems. The rank-based analysis is a highly efficient, attractive alternative to the traditional least squares ANOVA and covariance. The coefficients of determination for both classes of estimates are discussed in the chapter. These are robust analogues of the popular R 2 statistic based on LS estimates.

Finite sample stability properties of the least median of squares estimator

Abstract: This article examines the stability properties of the least median of squares (LMS) estimate. Attention is focussed on LMS since it is arguably the most widely used high breakdown regression estimate. The differing roles of breakdown point and influence function in producing estimates which are stable to changes in the data are discussed. Simulations and real examples are used to illustrate the extent to which the LMS estimate can change when small changes are made to centrally located data. It is shown that this instability is a consequence of the fact that the LMS estimate has an influence function which is unbounded to the effects of centrally located x's and is not merely a consequence of the exact fit property and the curse of dimensionality, as argued by Rousseeuw (1994).