A Review of Some Asymptotic Properties of Trimmed Sums of Multivariate Data

  • R. A. Maller
Part of the Progress in Probability book series (PRPR, volume 23)


‘Trimming’ in this article will be used to describe the idea of removing points of a sample (usually ‘extreme points’) in order to improve the properties of estimators based on the sample. We will also consider procedures in which sample points are weighted so that the influence of the extremes is reduced. The sample will always consist of n independent and identically distributed (iid) random vectors in ℝd, and our emphasis will be on the (asymptotic) behaviour of that portion of it which remains after deleting or downweighting the extremes, rather than in the behaviour of the extremes themselves. The object of interest will be the estimation of location or scale of the distribution of the Sample, or more precisely, in the ‘robust’ estimation of location and scale due to removal or downweighting of extremes. Thus we are led to the investigation of the asymptotic properties of ‘trimmed sums’ (or ‘trimmed means’ or ‘robust variance matrices’) in ℝd.


Convex Hull Order Statistic Asymptotic Normality Iterate Logarithm Minimum Covering 
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  1. ALDOUS, D.J., FRISTEDT, B., GRIFFIN, P.S., and PRUITT, W.E.(1991) The number of extreme points in the convex hull of a random sample (preprint).Google Scholar
  2. AROV, D.Z. AND BOBROV, A.A.(1960) The extreme terms of a sample and their role in the sum of independent variables. Theor. Prob. App l5, S77–396.MathSciNetGoogle Scholar
  3. BARNETT, V.D. AND LEWIS, T.(1984) Outliers in Statistical Data. 2nd Ed, Wiley, New York.MATHGoogle Scholar
  4. BHATTACHARYA, R.N.(1977) Refinements of the multidimensional central limit theorem and applications. Ann. Probab., 5 1–27.MATHCrossRefGoogle Scholar
  5. BICKEL, P.J.(1965) On some robust estimates of location. Ann. Math. Statist. 36, 847–848.MathSciNetMATHCrossRefGoogle Scholar
  6. CARNAL, H.(1970) Die konvexe Hulle von n rotationssynmietrisch verteilten Pimk- ten. Z. Wahrscheinlichkeitstheorie verw. Geb, 15, 168–179.MathSciNetMATHCrossRefGoogle Scholar
  7. COLLINS, J.R.(1976) Robust estimation of a location parameter in the presence of asymmetry. Ann. Statist\., 4, 68–85.MathSciNetMATHCrossRefGoogle Scholar
  8. COLLINS, J.R.(1982) Robust M-estimators of location vectors. J. Mult. Anal, 12, 480–492.MATHCrossRefGoogle Scholar
  9. CSÖRGÖ, M., CSÖRGÖ, S., HÖRVATH, L. AND MASON, D.M.(1986) Normal and stable convergence of integral functions of the empirical distribution function. Ann. Prob\., 14, 86–118.CrossRefGoogle Scholar
  10. CSÖRGÖ, S., HÖRVATH, L. AND MASON, D.M.(1986) What portion of a.sample makes a partial sum asymptotically stable or normal? Prob. Theor. Rel. Fields. 72, 1–16.CrossRefGoogle Scholar
  11. CSöRGő, S., HAEUSLER, E. AND MASON, D.M.(1988) The asymptotic distribution of trimmed sums. Ann. Prob\., 16, 672–699.CrossRefGoogle Scholar
  12. CSÖRGÖ, S., AND HÖRVATH, L.(1988) Asymptotic representations of self normalised simis. Prob, and Math. Statistics, 9.1, 15–24.Google Scholar
  13. CSÖRGÖ, S., HAEUSLER, E. AND MASON, D.M.(1990) A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables (Preprint).Google Scholar
  14. DANIELS, H.E.(1952) The covering circle of a sample from a circular normal distribution. Biometrika 39, 137–143.MathSciNetMATHGoogle Scholar
  15. DAVIES, P.L.(1987) Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices. Ann. Statist 15, 1269–1292.MathSciNetMATHCrossRefGoogle Scholar
  16. DAVIS, R., MULROW, E., AND RESNICK, S.(1987) The convex hull of a random sample in R. Stochastic Models 3, 1–29.MathSciNetMATHCrossRefGoogle Scholar
  17. DE HAAN, L. AND RESNICK, S.I.(1977) Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie verw. Geb\., 40, 317–337.MATHCrossRefGoogle Scholar
  18. DIACONIS, P. AND FREEDMAN, D.(1984) Asymptotics of graphical projection pursuit. Ann. Statist. 12, 793–815.MathSciNetMATHCrossRefGoogle Scholar
  19. DONOHO, D. AND HUBER, P.(1983) The notion of breakdown point. In: A Festschrift for Erich Lehmann, Eickel, Doksum Eds, (Wadsworth).Google Scholar
  20. EGOROV, V.A., AND NEVZOROV, V.B.(1981) On a rate of convergence to a normal law of sums of induced order statistics. Notes of the Science Seminars of LOMI, 108, 45–46.MathSciNetMATHGoogle Scholar
  21. FELLER, W. (1968) An extension of the law of the iterated logarithm to variables without variance. J. Math, Mech. 18, 343–355.MathSciNetMATHGoogle Scholar
  22. FRIEDMAN, J. AND TUKEY, J.W.(1974) A projection pursuit algorithm for exploratory data analysis. IEEE Transactions on Computers, C-23, 881–889.CrossRefGoogle Scholar
  23. GREEN, P.J.(1981) Peeling bivariate data. In: Interpreting Multivariate Data (ed. V Barnett) Wiley, 3–18.Google Scholar
  24. GRIFFIN, P.S.(1986) Matrix normalised sums of independent identically distributed random vectors. Ann. Prob\., 14, 224–246.MathSciNetMATHCrossRefGoogle Scholar
  25. GRIFFIN, P.S.(1989) Asymptotic normality of self-normalised sums, (preprint).Google Scholar
  26. GRIFFIN, P.S. AND PRUITT, W.E.(1987) The central limit problem for trimmed sums. Math. Proc. Camh. Phil. Soc, 102, 329–349.MathSciNetMATHCrossRefGoogle Scholar
  27. GRIFFIN, P.S. AND PRUITT, W.E.(1989) Asymptotic normality and subsequen- tial limits of trimmed sums. Ann. Prob. 17, 1186–1210.MathSciNetMATHCrossRefGoogle Scholar
  28. GRIFFIN, P.S. AND KUELBS, J.(1989a) Self normalised laws of the iterated logarithm. (Preprint).Google Scholar
  29. GRIFFIN, P.S. AND KUELBS, J.(1989b) Some extensions of the LIL via self normalisations Ann. Prob. 17, 1571–1601.MathSciNetMATHCrossRefGoogle Scholar
  30. GROENEBOOM, P.(1988) Limit theorems for convex hulls. Prob. Theor. Related Fields, 79, 327–368.MathSciNetMATHCrossRefGoogle Scholar
  31. GRUBEL, R.(1988) The length of the shorth. Ann. Statist, 16, 619–628.MathSciNetCrossRefGoogle Scholar
  32. HAEUSLER, E. AND MASON, D.M.(1987) Laws of the iterated logarithm for sums of the middle portion of the sample. Math. Proc. Camb. Phil. Soc\., 101, 301–312.MathSciNetCrossRefGoogle Scholar
  33. HAEUSLER, E. AND MASON, D.M.(1990) A law of the iterated logarithm for modulus trimming. (Preprint).Google Scholar
  34. HAEUSLER, E.(1990) Laws of the iterated logarithm for svims of order statistics from a distribution with a regularly varying upper tail. (Preprint).Google Scholar
  35. HAHN, M.G. AND KLASS, M.J.(1980) Matrix normahsation of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Proh\., 8, 262–280.MathSciNetMATHCrossRefGoogle Scholar
  36. HAHN, M.G. AND KLASS, M.J.(1981) The multidimensional central limit theorem for arrays normed by affine transformations. Ann. Prob\., 9, 611–623.MathSciNetMATHCrossRefGoogle Scholar
  37. HAHN, M.G., KUELBS, J. AND SAMUR, J.D.(1987) Asymptotic normality of trimmed sums of (mixing random variables. Ann. Prob\., 15, 1395–1418.MathSciNetMATHCrossRefGoogle Scholar
  38. HAHN, M.G., AND KUELBS, J.(1989) Asymptotic normality and the LIL for trinamed sums: the general case. J. Theoret. Prob. 3, 137–168.MathSciNetCrossRefGoogle Scholar
  39. HAHN, M.G., KUELBS, J. AND WEINER, D.C.(1989a) The asymptotic joint distribution of self normalised censored sums and sums-of-squares. (Preprint).Google Scholar
  40. HAHN, M.G., KUELBS, J. AND WEINER, D.C.(1989b) The asymptotic distribution of magnitude Winsorised sums via self-normalisation. J. Theor. Prob\., 3, 137–168.MathSciNetCrossRefGoogle Scholar
  41. HALL, P.(1984). On the influence of extremes on the rate of convergence in the central limit theorem. Ann. Prob\., 12, 154–172.MATHCrossRefGoogle Scholar
  42. HAMPEL, F.R., ROUSSEEUW, P.J., RONCHETTI, E.M. AND STAHEL, W.A. (1986).Robust Statistics- the Approach based on Influence Functions. Wiley, New York.MATHGoogle Scholar
  43. HUBER, P.J.(1985) Projection Pursuit. Ann. Statist, 13, 435–522.MathSciNetMATHCrossRefGoogle Scholar
  44. HUBER, P.J.(1970) Studentizing robust estimates. In: Nonparametric Techniques in Statistical Inference, M. L. Puri., Cambridge Univ. Press.Google Scholar
  45. JAECKEL, L.A.(1971) Some flexible estimates of location. Ann. Math. Statist\., 42, 1540–1552.MathSciNetMATHCrossRefGoogle Scholar
  46. JOHNSTONE, I.(1987) Discussion to: Jones, M.C., and Sibson, R.: What is projection pursuit? J. R. Statist. Soc\., 150, 1–36.Google Scholar
  47. JONES, M.C. AND SIBSON, R (1987) What is projection pursuit? J. R. Statist. Soc. A, 150 1–36.MathSciNetMATHCrossRefGoogle Scholar
  48. MALLER, R.A.(1981) A theorem on products of random variables, with apphcation to regression. Aust. J. Statist, 23, 25–37.MathSciNetGoogle Scholar
  49. MALLER, R. A.(1982) Asymptotic normality of lightly trimmed means - a converse. Math. Proc. Camh. Phil. Soc\., 92, 535–545.MathSciNetMATHCrossRefGoogle Scholar
  50. MALLER, R.A.(1984) Relative stability of trimmed sums.Z.Wahrscheinlichkeits- theorie verw. Geb\., 66, 61–80.MathSciNetMATHCrossRefGoogle Scholar
  51. MALLER, R.A.(1988a) Asymptotic normality of trimmed means in higher dimensions. Ann. Prob\., 16, 1608–1622.MathSciNetMATHCrossRefGoogle Scholar
  52. MALLER, R.A.(1988b) A functional law of the iterated logarithm for distributions in the domain of partial attraction of the normal distribution. Stock. Proc. AppL, 27, 179–194.MathSciNetMATHCrossRefGoogle Scholar
  53. MALLER, R.A.(1990a) Some consistency results on projection pursuit estimators of location and scale. Canad. J. Statist, 17, 81–90.MathSciNetGoogle Scholar
  54. MALLER, R.A.(1990b) Defining extremes and trimming by minimum covering sets. Stock. Proc. Appl. 35.Google Scholar
  55. MARONNA, R.A. (1976) Robust M-estimators of multivariate location and scatter. Ann. Statist. 1, 51–67.MathSciNetCrossRefGoogle Scholar
  56. MASON, D.M. (1982a) Laws of large numbers for siims of extreme values. Ann. Prob. 10, 754–764.MATHCrossRefGoogle Scholar
  57. MASON, D.M. (1982b) Some characterisations of strong laws for linear functions of order statistics. Ann. Prob. 10, 1051–1057.MATHCrossRefGoogle Scholar
  58. MASON, D.M. AND SHORACK, G.R.(1990) Necessary and sufficient conditions for asymptotic normality of L-statistics. (preprint).Google Scholar
  59. MATHERON, G.(1975) Random Sets and Integral Geometry, Wiley, New York.MATHGoogle Scholar
  60. MORI, T.(1976) The strong law of large nimibers when extreme terms are excluded from sums. Z. Wahrscheinlickkeitstkeorie verw. Geb\., 36, 189–194.MATHCrossRefGoogle Scholar
  61. MORI, T.(1977) Stability for sums of iid random variables when extreme terms are excluded. Z. Wahrscheinlichkeitstheorie verw. Geh. , 40, 159–167MATHCrossRefGoogle Scholar
  62. MORL T.(1981) The relation of sums and extremes of random variables. Session similarly booklet: invited papers, Buenos-Aires Session, Nov 30-Dec 11, 1981 (International Statistical Institute).Google Scholar
  63. MORI, T.(1984) On the limit distributions of lightly trimmed sums. Math. Proc. Camh. Phil. Soc\., 96, 507–516.MATHCrossRefGoogle Scholar
  64. PRUITT, W.E.(1988) Sums of independent random variables with the extreme terms excluded. In: Probability and Statistics, Essays in honour of Franklin A. Graybill, J. N. Srivastava, Ed, North Holland.Google Scholar
  65. RAYNAUD, H.(1970) Sur I’envelope convexe des nuages de points aleatoires dans R n . J. Appl. Prob. 7, 35–48.MathSciNetMATHCrossRefGoogle Scholar
  66. RESNICK, S. (1988)Association and multivariate extreme value distributions. In: Studies in Modelling and Statistical Science C.C. Heyde, Ed. Aust. J. Statist. 30A, 261–271.Google Scholar
  67. ROUSSEEUW, X (1986) Multivariate estimation with high breakdown point. In: Mathematical Statistics and Its Applications, Grossman, Vincze and Wertz, Eds, Reidal, Dordrecht, 283–297.Google Scholar
  68. RUPPERT, D. AND CARROLL, R.J.(1980) Trimmed least squares estimation in the linear model. J. Amer. Statist. Assoc\., 75, 828–297.MathSciNetMATHCrossRefGoogle Scholar
  69. SATO, K.(1973) A note on infinitely divisible distributions and their Levy measures. Sci. Rep. Tokyo Kyoiku Daigaku Sect A, 12, 101–109.MathSciNetMATHGoogle Scholar
  70. SHORACK, G.(1974) Random means. Ann. Statist, 2, 661–675.MathSciNetMATHCrossRefGoogle Scholar
  71. SHORACK, G.R. AND WELLNER, J.A.(1985) Empirical Processes with Application to Statistics. Wiley, N.Y.Google Scholar
  72. SIBSON, R.(1972) Discussion of a paper by H.R Wynn. J. Roy Statist. Soc. , B 34, 181–183.Google Scholar
  73. SILVERMAN, B.W. AND TITTERINGTON, D.M.(1980) Minimum covering ellipses. Siam J. Sci. Statist. Comput, 1, 401–409.MathSciNetMATHCrossRefGoogle Scholar
  74. STIGLER, S.M.(1973) The asymptotic distribution of the trimmed mean.Ann. Statist\., 1, 472–477.MathSciNetMATHCrossRefGoogle Scholar
  75. SWEETING, T.J.(1977) Speeds of convergence for the multidimensional central limit theorem. Ann. Prob\., 5, 28–41.MathSciNetMATHCrossRefGoogle Scholar
  76. TITTERINGTON, D.M.(1975) Optimal design: some geometrical aspects of D- optimality. Biometrika, 62, 313–320.MathSciNetMATHGoogle Scholar
  77. TITTERINGTON, D.M.(1978) Estimation of correlation coefficients by ellipsoidal trimming. Appl. Statist\., 27, 227–234.MATHCrossRefGoogle Scholar
  78. TUKEY, J.W.(1947) Nonparametric estimation II. Statistically equivalent blocks and tolerance regions in the continuous case. Ann Math. Statist\., 18, 529–539.MathSciNetMATHCrossRefGoogle Scholar
  79. TYLER, D.E.(1981) Asymptotic inference for eigenvectors. Ann Statist. , 9, 725–736.MathSciNetMATHCrossRefGoogle Scholar
  80. WALD, A.(1943) An extension of Wilks’ method for setting tolerance limits. Ann. Math. Statist, 14, 45–55.MathSciNetMATHCrossRefGoogle Scholar
  81. WELSH, A.H.(1987) The trimmed mean in the linear model. Ann. Statist., 15, 20–36.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • R. A. Maller
    • 1
  1. 1.Department of MathematicsThe University of Western AustraliaNedlandsAustralia

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