## The Annals of Statistics

### Multidimensional trimming based on projection depth

Yijun Zuo

#### Abstract

As estimators of location parameters, univariate trimmed means are well known for their robustness and efficiency. They can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy-tailed models. This paper introduces multidimensional trimmed means based on projection depth induced regions. Robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point. The influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators. It is found that the projection depth trimmed means are highly robust locally as well as globally. Asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions. The strong consistency, asymptotic representation and limiting distribution of the depth trimmed means are obtained. Relative to the mean and other leading competitors, the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated. Simulation studies confirm the validity of the asymptotic efficiency results at finite samples.

#### Article information

Source
Ann. Statist., Volume 34, Number 5 (2006), 2211-2251.

Dates
First available in Project Euclid: 23 January 2007

https://projecteuclid.org/euclid.aos/1169571795

Digital Object Identifier
doi:10.1214/009053606000000713

Mathematical Reviews number (MathSciNet)
MR2291498

Zentralblatt MATH identifier
1106.62057

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G20: Asymptotic properties 62G35: Robustness

#### Citation

Zuo, Yijun. Multidimensional trimming based on projection depth. Ann. Statist. 34 (2006), no. 5, 2211--2251. doi:10.1214/009053606000000713. https://projecteuclid.org/euclid.aos/1169571795

#### References

• Bai, Z.-D. and He, X. (1999). Asymptotic distrbutions of the maximal depth estimators for regression and multivariate location. Ann. Statist. 27 1616--1637.
• Bickel, P. J. (1965). On some robust estimates of location. Ann. Math. Statist. 36 847--858.
• Bickel, P. J. and Lehmann, E. L. (1975). Descriptive statistics for nonparametric models. II. Location. Ann. Statist. 3 1045--1069.
• Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statistics, Harvard Univ.
• Donoho, D. L. and Gasko, M. (1987). Multivariate generalizations of the median and trimmed mean. I. Technical Report 133, Dept. Statistics, Univ. California, Berkeley.
• Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803--1827.
• 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.
• Dümbgen, L. (1992). Limit theorem for the simplicial depth. Statist. Probab. Lett. 14 119--128.
• 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.
• Hampel, F. R., Ronchetti, E. Z., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.
• Huber, P. J. (1972). Robust statistics: A review. Ann. Math. Statist. 43 1041--1067.
• Huber, P. J. (1981). Robust Statistics. Wiley, New York.
• Kim, S. (1992). The metrically trimmed mean as a robust estimator of location. Ann. Statist. 20 1534--1547.
• Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405--414.
• 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.
• 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.
• Lopuhaä, H. P. and Rousseeuw, P. J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19 229--248.
• Massé, J.-C. (2004). Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean. Bernoulli 10 397--419.
• Nolan, D. (1992). Asymptotics for multivariate trimming. Stochastic Process. Appl. 42 157--169.
• Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
• Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
• Ranga Rao, R. (1962). Relation between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33 659--680.
• Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
• Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
• Stahel, W. A. (1981). Breakdown of covariance estimators. Research Report 31, Fachgruppe für Statistik, ETH, Zürich.
• Stigler, S. M. (1973). The asymptotic distribution of the trimmed mean. Ann. Statist. 1 472--477.
• Stigler, S. M. (1977). Do robust estimators work with real data (with discussion)? Ann. Statist. 5 1055--1098.
• Tukey, J. W. (1948). Some elementary problems of importance to small sample practice. Human Biology 20 205--214.
• Tukey, J. W. (1975). Mathematics and the picturing of data. In Proc. International Congress of Mathematicians 2 523--531. Canadian Mathematical Congress, Montreal.
• Tukey, J. W. and McLaughlin, D. H. (1963). Less vulnerable confidence and significance procedures for location based on a single sample: Trimming/Winsorization. I. Sankhyā Ser. A 25 331--352.
• Tyler, D. E. (1994). Finite sample breakdown points of projection based multivariate location and scatter statistics. Ann. Statist. 22 1024--1044.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, Berlin.
• Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist. 31 1460--1490.
• Zuo, Y. (2004). Robustness of weighted $L^p$-depth and $L^p$-median. Allg. Stat. Arch. 88 215--234.
• Zuo, Y., Cui, H. and He, X. (2004). On the Stahel--Donoho estimator and depth-weighted means of multivariate data. Ann. Statist. 32 167--188.
• Zuo, Y., Cui, H. and Young, D. (2004). Influence function and maximum bias of projection depth based estimators. Ann. Statist. 32 189--218.
• Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist. 28 461--482.