Bai, Zhi-Dong; He, Xuming Asymptotic distributions of the maximal depth estimators for regression and multivariate location. (English) Zbl 1007.62009 Ann. Stat. 27, No. 5, 1616-1637 (1999). Summary: We derive the asymptotic distribution of the maximal depth regression estimator proposed by P.J. Rousseeuw and M. Hubert [J. Am. Stat. Assoc. 94, No. 446, 388-433 (1999; this Zbl 1007.62060)]. The estimator is obtained by maximizing a projection-based depth and the limiting distribution is characterized through a max-min operation of a continuous process. The same techniques can be used to obtain the limiting distribution of some other depth estimators including Tukey’s deepest point based on half-space depth. Results for the special case of two-dimensional problems have been available, but the earlier arguments have relied on some special geometric properties in the low-dimensional space. This paper completes the extension to higher dimensions for both regression and multivariate location models. Cited in 4 ReviewsCited in 45 Documents MSC: 62E20 Asymptotic distribution theory in statistics 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis Citations:Zbl 1007.62060 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Donoho, D. L. and Gasko, M. (1992). Breakdown properties oflocation estimates based on half space depth and projection outlyingness. Ann. Statist. 20 1803-1827. · Zbl 0776.62031 · doi:10.1214/aos/1176348890 [2] He, X. (1999). Comment on ”Regression depth,” by P. J. Rousseeuw and M. Hubert. J. Amer. Statist. Assoc. 94 403-404. JSTOR: · Zbl 1007.62060 · doi:10.2307/2670155 [3] He, X., Jureckova, J., Koenker, R. and Portnoy, S. (1990). Tail behavior ofregression estimators and their breakdown points. Econometrica 58 1195-1214. JSTOR: · Zbl 0745.62030 · doi:10.2307/2938306 [4] He, X. and Portnoy, S. (1998). Asymptotics ofthe deepest line. In Applied Statistical Science III: Nonparametric Statistics and Related Topics (S.E. Ahmed, M. Ahsanullah, and B.K. Sinha, eds.) 71-81. Nova Science Publications, New York. [5] He, X. and Shao, Q. M. (1996). A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs. Ann. Statist. 24 2608-2630. · Zbl 0867.62012 · doi:10.1214/aos/1032181172 [6] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063 · doi:10.1214/aos/1176347498 [7] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. JSTOR: · Zbl 0373.62038 · doi:10.2307/1913643 [8] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: descriptive statistics, graphics and inference. Ann. Statist. 27. 783-840. · Zbl 0984.62037 · doi:10.1214/aos/1018031260 [9] Lo eve, M. (1977). Probability Theory, 4th ed. Springer, New York. · Zbl 0095.12201 [10] Nolan, D. (1999). On Min-Max Majority and Deepest Points. Statist. Probab. Lett. 43 325-334. · Zbl 0947.62024 · doi:10.1016/S0167-7152(98)00173-4 [11] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045 [12] Rousseeuw, P. J. and Hubert, M. (1999). Regression depth (with discussions). J. Amer. Statist. Assoc. 94 388-402. JSTOR: · Zbl 1007.62060 · doi:10.2307/2670155 [13] Rousseeuw, P. J. and Struyf, A. (1998). Computing location depth and regression depth in higher dimensions. Statist. Comput. 8 193-203. [14] Tukey, J. W. (1975). Mathematics and the picturing ofdata. In Proceedings of the International Congress of Mathematicians, Vancouver 2 523-531. · Zbl 0347.62002 [15] Tyler, D. E. (1994). Finite sample breakdown points ofprojection based multivariate location and scatter statistics. Ann. Statist. 22 1024-1044. · Zbl 0815.62015 · doi:10.1214/aos/1176325510 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.