Generalized Radius Processes for Elliptically Contoured Distributions

The use of Mahalanobis distances has a long history in statistics. Given a sample of size n and general location and scatter estimators, mn and Σn, we can define “generalized” radii as . If we wish to trim observations based on the estimators mn and Σn, then it is natural to first remove the most remote ones (i.e., those with the largest 's). With this in mind, we define a process that maps the trimming proportion, α in (0, 1], to the generalized radius of the observation that has just been removed by this level of trimming. We analyze the asymptotic behavior of this process for elliptically contoured distributions. We show that the limit law depends only on the elliptical family considered and how Σn serves to estimate the underlying “scale” factor through its determinant. We carry out Monte Carlo simulations for finite sample sizes, and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded.

[1]  C. Croux,et al.  Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator , 1999 .

[2]  David L. Woodruff,et al.  Identification of Outliers in Multivariate Data , 1996 .

[3]  Daniel J. Mundfrom,et al.  On Using Asymptotic Critical Values in Testing for Multivariate Normality , 2002 .

[4]  P. Mahalanobis.  On the generalized distance in statistics , 1936 .

[5]  Douglas M. Hawkins.  Identification of Outliers , 1980, Monographs on Applied Probability and Statistics.

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

[7]  M. Healy,et al.  Multivariate Normal Plotting , 1968 .

[8]  C. Croux,et al.  Principal Component Analysis Based on Robust Estimators of the Covariance or Correlation Matrix: Influence Functions and Efficiencies , 2000 .

[9]  D. Hawkins.  Multivariate Statistics: A Practical Approach , 1990 .

[10]  P. Rousseeuw,et al.  Unmasking Multivariate Outliers and Leverage Points , 1990 .

[11]  James A. Koziol,et al.  A class of invariant procedures for assessing multivariate normality , 1982 .

[12]  G. Willems,et al.  A robust Hotelling test , 2002 .

[13]  Maria L. Rizzo,et al.  A new test for multivariate normality , 2005 .

[14]  H. Riedwyl,et al.  Multivariate Statistics: A Practical Approach , 1988 .

[15]  A. Hadi,et al.  BACON: blocked adaptive computationally efficient outlier nominators , 2000 .

[16]  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 .

[17]  J. Schmee.  An Introduction to Multivariate Statistical Analysis , 1986 .

[18]  Adrian Baddeley,et al.  Integrals on a moving manifold and geometrical probability , 1977, Advances in Applied Probability.

[19]  Andrzej S. Kosinski,et al.  A procedure for the detection of multivariate outliers , 1998 .

[20]  P. L. Davies,et al.  Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices , 1987 .

[21]  A. Afifi,et al.  On Tests for Multivariate Normality , 1973 .

[22]  Douglas M. Hawkins,et al.  A new test for multivariate normality and homoscedasticity , 1981 .

[23]  A. Gordaliza,et al.  Robustness Properties of k Means and Trimmed k Means , 1999 .

[24]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[25]  Anthony C. Atkinson,et al.  Exploring Multivariate Data with the Forward Search , 2004 .

[26]  M. Jhun,et al.  Asymptotics for the minimum covariance determinant estimator , 1993 .

[27]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[28]  A. Hadi.  Identifying Multiple Outliers in Multivariate Data , 1992 .

[29]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[30]  Geoffrey S. Watson,et al.  Distribution Theory for Tests Based on the Sample Distribution Function , 1973 .

[31]  P. Rousseeuw.  Multivariate estimation with high breakdown point , 1985 .

[32]  Sándor Csörgő,et al.  Consistency of some tests for multivariate normality , 1989 .

[33]  Bell Telephone,et al.  ROBUST ESTIMATES, RESIDUALS, AND OUTLIER DETECTION WITH MULTIRESPONSE DATA , 1972 .

[34]  David E. Tyler,et al.  On the uniqueness of S-functionals and M-functionals under nonelliptical distributions , 2000 .

[35]  Graphical Detection of Regression Outliers and Mixtures , 1999 .

[36]  J. Durbin.  Distribution theory for tests based on the sample distribution function , 1973 .