A Monte Carlo study of the accuracy and robustness of ten bivariate location estimators

In a Monte Carlo study, ten bivariate location estimators are compared as regards their accuracy and robustness. In addition to the arithmetic mean, five bivariate medians and four depth-based trimmed means are thus investigated. The behavior of the estimators is examined under various sampling situations determined by three sample sizes and 26 underlying distributions, 14 of which are centrally symmetric and 12 are asymmetric contaminated normals. Performance is assessed through numerical functions of the sample mean squared error and bias matrices.

[1]  H. Oja,et al.  The Oja Bivariate Median , 1992 .

[2]  P. Chaudhuri.  On a geometric notion of quantiles for multivariate data , 1996 .

[3]  C. J. Lawrence.  Robust estimates of location : survey and advances , 1975 .

[4]  Frederick R. Forst,et al.  On robust estimation of the location parameter , 1980 .

[5]  H. Oja,et al.  Asymptotic Properties of the Generalized Median in the Case of Multivariate Normality , 1985 .

[6]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[7]  W. R. Buckland,et al.  Contributions to Probability and Statistics , 1960 .

[8]  D. F. Andrews,et al.  Robust Estimates of Location: Survey and Advances. , 1975 .

[9]  P. Rousseeuw,et al.  Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices , 1991 .

[10]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[11]  Peter J. Bickel,et al.  On Some Alternative Estimates for Shift in the $P$-Variate One Sample Problem , 1964 .

[12]  Jean Meloche,et al.  Multivariate L-estimation , 1999 .

[13]  C. Small.  A Survey of Multidimensional Medians , 1990 .

[14]  P. J. Huber.  The 1972 Wald Lecture Robust Statistics: A Review , 1972 .

[15]  P. Chaudhuri,et al.  A note on the robustness of multivariate medians , 1999 .

[16]  T. Hettmansperger,et al.  Robust Nonparametric Statistical Methods , 1998 .

[17]  Carlos Antonio LeOan,et al.  La médiane simpliciale d'Oja: Existence, unicité et stabilité , 1993 .

[18]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

[19]  J. Haldane.  Note on the median of a multivariate distribution , 1948 .

[20]  F. Bedall,et al.  Algorithm AS 143: The Mediancentre , 1979 .

[21]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[22]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[23]  P. Rousseeuw,et al.  Constructing the bivariate Tukey median , 1998 .

[24]  P. Rousseeuw,et al.  Bivariate location depth , 1996 .

[25]  P. Rousseeuw,et al.  High-dimensional computation of the deepest location , 2000 .

[26]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[27]  D. Donoho,et al.  Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness , 1992 .

[28]  Peter Rousseeuw,et al.  Computing location depth and regression depth in higher dimensions , 1998, Stat. Comput..

[29]  Roland P. Falkner,et al.  History of statistics , 1891 .

[30]  P. Rousseeuw,et al.  The depth function of a population distribution , 1999, Metrika.

[31]  M. E. Johnson,et al.  Multivariate Statistical Simulation , 1988 .

[32]  H. Oja.  Descriptive Statistics for Multivariate Distributions , 1983 .

[33]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

[34]  D. Ruppert.  Robust Statistics: The Approach Based on Influence Functions , 1987 .