Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Strong laws for generalized absolute Lorenz curves when data are stationary and ergodic sequences


Authors: Roelof Helmers and Ricardas Zitikis
Journal: Proc. Amer. Math. Soc. 133 (2005), 3703-3712
MSC (2000): Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-05-08096-2
Published electronically: June 28, 2005
MathSciNet review: 2163610
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider generalized absolute Lorenz curves that include, as special cases, classical and generalized $L$-statistics as well as absolute or, in other words, generalized Lorenz curves. The curves are based on strictly stationary and ergodic sequences of random variables. Most of the previous results were obtained under the additional assumption that the sequences are weakly Bernoullian or, in other words, absolutely regular. We also argue that the latter assumption can be undesirable from the applications point of view.


References [Enhancements On Off] (What's this?)

  • [1996] AARONSON, J., BURTON, R., DEHLING, H., GILAT, D., HILL, T., WEISS, B. (1996). Strong laws for $L$- and $U$-statistics. Trans. Amer. Math. Soc. 348, 2845-2866. MR 1363941 (97b:60047)
  • [1998] ARCONES, M. A. (1998). The law of large numbers for $U$-statistics under absolute regularity. Electron. Comm. Probab. 3, 13-19. MR 1624866 (99d:60038)
  • [1986] BERBEE, H. (1986). Periodicity and absolute regularity. Israel J. Math. 55, 289-304. MR 0876396 (88b:60088)
  • [1981] BICKEL, P. J. AND FREEDMAN, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196-1217. MR 0630103 (83a:62051)
  • [1968] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. MR 0233396 (38:1718)
  • [1999] BOROVKOVA, S., BURTON, R., AND DEHLING, H. (1999). Consistency of the Takens estimator for the correlation dimension. Ann. Appl. Probab. 9, 376-390. MR 1687339 (2000g:60035)
  • [2002] DAVYDOV, Y. AND ZITIKIS, R. (2002). Convergence of generalized Lorenz curves based on stationary ergodic random sequences with deterministic noise. Statist. Probab. Lett. 59, 329-340. MR 1935667 (2003j:60045)
  • [2003] DAVYDOV, Y. AND ZITIKIS, R. (2003). Generalized Lorenz curves and convexifications of stochastic processes. J. Appl. Probab. 40, 906-925. MR 2012676 (2004g:60059)
  • [2004a] DAVYDOV, Y. AND ZITIKIS, R. (2004a). The influence of deterministic noise on empirical measures generated by stationary processes. Proc. Amer. Math. Soc. 132, 1203-1210. MR 2045439 (2005e:60080)
  • [2004b] DAVYDOV, Y. AND ZITIKIS, R. (2004b). Convex rearrangements of random elements. In: Asymptotic Methods in Stochastics (eds. L. Horváth and B. Szyszkowicz), pp. 141-171, Volume 44, Fields Institute Communications, American Mathematical Society, Providence, RI. MR 2106853
  • [1997] GILAT, D. AND HELMERS, R. (1997). On strong laws for generalized $L$-statistics with dependent data. Comment. Math. Univ. Carolin. 38, 187-192. MR 1455483 (98g:62096)
  • [1912] GINI, C. (1912). Variabilitá e mutabilita. Reprinted in: Memorie di metodologia statistica. (Ed. E. Pizetti and T. Salvemini; 1955) Libreria Eredi Virgilio Veschi, Rome.
  • [1977] GOLDIE, C. M. (1977). Convergence theorems for empirical Lorenz curves and their inverses. Advances in Appl. Probability 9, 765-791. MR 0478267 (57:17752)
  • [1982] HELMERS, R. (1982). Edgeworth Expansions for Linear Combinations of Order Statistics. Mathematical Centre Tracts, 105. Mathematisch Centrum, Amsterdam. MR 0665747 (84f:62029)
  • [1988] HELMERS, R., JANSSEN, P. AND SERFLING, R. (1988). Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized $L$-statistics. Probab. Theory Related Fields 79, 75-93. MR 0952995 (89h:60046)
  • [1905] LORENZ, M. O. (1905). Methods for measuring the concentration of wealth. Amer. Stat. Assoc. 9, 209-219.
  • [1991] ROSENBLATT, M. (1991). Stochastic Curve Estimation. Institute of Mathematical Statistics, Hayward, CA.
  • [1980] SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. MR 0595165 (82a:62003)
  • [1984] SERFLING, R. J. (1984). Generalized $L$-, $M$-, and $R$-statistics. Ann. Statist. 12, 76-86. MR 0733500 (85i:62018)
  • [1994] SHALIT, H. AND S. YITZHAKI, S. (1994). Marginal conditional stochastic dominance. Management Science 40, 670-684.
  • [1986] SHORACK, G. R. AND WELLNER, J. A. (1986). Empirical Processes With Applications to Statistics. Wiley, New York. MR 0838963 (88e:60002)
  • [1983] SHORROCKS, A. J. (1983). Ranking income distributions. Economica 50, 3-17.
  • [1980] VAN ZWET, W. R. (1980). A strong law for linear functions of order statistics. Ann. Probab. 8, 986-990. MR 0586781 (81j:60043)
  • [1991] YITZHAKI, S. AND OLKIN, I. (1991). Concentration indices and concentration curves. In: Stochastic Orders and Decision under Risk (Hamburg, 1989), pp. 380-392, IMS Lecture Notes Monogr. Ser., 19, Inst. Math. Statist., Hayward, CA. MR 1196066 (93g:90022)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60F15

Retrieve articles in all journals with MSC (2000): 60F15


Additional Information

Roelof Helmers
Affiliation: Centre for Mathematics and Computer Science (CWI), Kruislaan 413, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: R.Helmers@cwi.nl

Ricardas Zitikis
Affiliation: Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada N6A 5B7
Email: zitikis@stats.uwo.ca

DOI: https://doi.org/10.1090/S0002-9939-05-08096-2
Received by editor(s): July 6, 2004
Published electronically: June 28, 2005
Additional Notes: The second author was partially supported by the Netherlands Organization for Scientific Research (NWO), as well as by a Discovery Research Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society