## Projection-based depth functions and associated medians.(English)Zbl 1046.62056

Summary: A class of projection-based depth functions is introduced and studied. These projection-based depth functions possess desirable properties of statistical depth functions and their sample versions possess strong and order $$\sqrt n$$ uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained.
Affine equivariant multivariate medians induced from projection-based depth functions are probed. The limiting distributions as well as the strong and order $$\sqrt n$$ consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of a leading depth-induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that, with appropriate choices of univariate location and scale estimators, the projection medians have a very high finite sample breakdown point and relative efficiency, much higher than those of the halfspace median.
Based on the results obtained, it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators, respectively.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62F35 Robustness and adaptive procedures (parametric inference) 62G05 Nonparametric estimation 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics
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### References:

 [1] Arcones, M. A. and Giné, E. (1993). Limit theorems for $$U$$-processes. Ann. Probab. 21 1494–1542. JSTOR: · Zbl 0789.60031 [2] Bai, Z.-D. and He, X. (1999). Asymptotic distributions of the maximal depth estimators for regression and multivariate location. Ann. Statist. 27 1616–1637. · Zbl 1007.62009 [3] Beran, R. J. and Millar, P. W. (1997). Multivariate symmetry models. In Festschrift for Lucien Le Cam : Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. L. Yang, eds.) 13–42. Springer, Berlin. · Zbl 0948.62039 [4] Cui, H. I. and Tian, Y. B. (1994). On the median absolute deviation of projected distribution and its applications. J. Systems Sci. Math. Sci. 14 63–72 (in Chinese). · Zbl 0826.62048 [5] Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statistics, Harvard Univ. [6] Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803–1827. JSTOR: · Zbl 0776.62031 [7] Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157–184. Wadsworth, Belmont, CA. · Zbl 0523.62032 [8] Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions . Chapman and Hall, London. · Zbl 0699.62048 [9] Gather, U. and Hilker, T. (1997). A note on Tyler’s modification of the MAD for the Stahel–Donoho estimator. Ann. Statist. 25 2024–2026. · Zbl 0881.62033 [10] Hall, P. and Welsh, A. H. (1985). Limit theorems for the median deviation. Ann. Inst. Statist. Math. 37 27–36. · Zbl 0591.62028 [11] He, X. and Wang, G. (1997). Convergence of depth contours for multivariate datasets. Ann. Statist. 25 495–504. · Zbl 0873.62053 [12] Huber, P. J. (1981). Robust Statistics . Wiley, New York. · Zbl 0536.62025 [13] Jurečková, J. and Sen, P. K. (1996). Robust Statistical Procedures : Asymptotics and Interrelations . Wiley, New York. · Zbl 0862.62032 [14] Kim, J. (2000). Rate of convergence of depth contours: With application to a multivariate metrically trimmed mean. Statist. Probab. Lett. 49 393–400. · Zbl 1146.62326 [15] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219. JSTOR: · Zbl 0703.62063 [16] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405–414. JSTOR: · Zbl 0701.62063 [17] Liu, R. Y. (1992). Data depth and multivariate rank tests. In $$L_1$$-Statistical Analysis and Related Methods (Y. Dodge, ed.) 279–294. North-Holland, Amsterdam. [18] Liu, R. Y. (1995). Control charts for multivariate processes. J. Amer. Statist. Assoc. 90 1380–1387. · Zbl 0868.62075 [19] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). Ann. Statist. 27 783–858. · Zbl 0984.62037 [20] Liu, R. Y. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. J. Amer. Statist. Assoc. 88 252–260. · Zbl 0772.62031 [21] Liu, R. Y. and Singh, K. (1997). Notions of limiting $$P$$-values based on data depth and bootstrap. J. Amer. Statist. Assoc. 92 266–277. · Zbl 0889.62010 [22] Lopuhaä, H. P. and Rousseeuw, J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19 229–248. JSTOR: · Zbl 0733.62058 [23] Massé, J. C. (1999). Asymptotics for the Tukey depth. [24] Massé, J. C. and Theodorescu, R. (1994). Halfplane trimming for bivariate distributions. J. Multivariate Anal. 48 188–202. · Zbl 0790.60024 [25] Mosteller, F. and Tukey, J. W. (1977). Data Analysis and Regression. Addison-Wesley, Reading, MA. [26] Nolan, D. (1992). Asymptotics for multivariate trimming. Stochastic Process. Appl. 42 157–169. · Zbl 0763.62007 [27] Nolan, D. (1999). On min-max majority and deepest points. Statist. Probab. Lett. 43 325–334. · Zbl 0947.62024 [28] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, Berlin. · Zbl 0544.60045 [29] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1996). Numerical Recipes in FORTRAN 90. The Art of Parallel Scientific Computing . Cambridge Univ. Press. · Zbl 0892.65001 [30] Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263–1268. · Zbl 1009.62047 [31] Randles, R. H. and Wolfe, D. A. (1979). Introduction to the Theory of Nonparametric Statistics . Wiley, New York. · Zbl 0529.62035 [32] Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871–880. · Zbl 0547.62046 [33] Rousseeuw, P. J. and Hubert, M. (1999). Regression depth (with discussion). J. Amer. Statist. Assoc. 94 388–433. · Zbl 1007.62060 [34] Rousseeuw, P. J. and Ruts, I. (1998). Constructing the bivariate Tukey median. Statist. Sinica 8 827–839. · Zbl 0905.62029 [35] Rousseeuw, P. J. and Ruts, I. (1999). The depth function of a population distribution. Metrika 49 213–244. · Zbl 1093.62540 [36] Rudin, W. (1987). Real and Complex Analysis , 3rd ed. McGraw-Hill, New York. · Zbl 0925.00005 [37] Serfling, R. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002 [38] Serfling, R. (2002a). Quantile functions for multivariate analysis: Approaches and applications. Statist. Neerlandica 56 214–232. · Zbl 1076.62054 [39] Serfling, R. (2002b). Generalized quantile processes based on multivariate depth functions, with applications in nonparametric multivariate analysis. J. Multivariate Anal . 83 232–247. · Zbl 1146.62327 [40] Stahel, W. A. (1981). Breakdown of covariance estimators. Research Report 31, Fachgruppe für Statistik, ETH, Zürich. [41] Struyf, A. and Rousseeuw, P. J. (2000). High-dimensional computation of the deepest location. Comput. Statist. Data Anal. 34 415–426. · Zbl 1046.62055 [42] Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians 523–531. Canad. Math. Congress, Montreal. · Zbl 0347.62002 [43] Tyler, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22 1024–1044. JSTOR: · Zbl 0815.62015 [44] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics . Springer, New York. · Zbl 0862.60002 [45] Zhang, J. (2002). Some extensions of Tukey’s depth function. J. Multivariate Anal. 82 134–165. · Zbl 1010.62058 [46] Zuo, Y., Cui, H. and He, X. (2001). On the Stahel–Donoho estimators and depth-weighted means of multivariate data. Ann. Statist. · Zbl 1105.62349 [47] Zuo, Y. and Serfling, R. (2000a). General notions of statistical depth function. Ann. Statist. 28 461–482. · Zbl 1106.62334 [48] Zuo, Y. and Serfling, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Ann. Statist. 28 483–499. · Zbl 1105.62343 [49] Zuo, Y. and Serfling, R. (2000c). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 55–79. · Zbl 1131.62305 [50] Zuo, Y. and Serfling, R. (2000d). Nonparametric notions of multivariate “scatter measure” and “more scattered” based on statistical depth functions. J. Multivariate Anal. 75 62–78. · Zbl 1011.62054
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