Strong laws for generalized absolute Lorenz curves when data are stationary and ergodic sequences
HTML articles powered by AMS MathViewer
- by Roelof Helmers and Ričardas Zitikis PDF
- Proc. Amer. Math. Soc. 133 (2005), 3703-3712 Request permission
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
- J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill, and B. Weiss, Strong laws for $L$- and $U$-statistics, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2845–2866. MR 1363941, DOI 10.1090/S0002-9947-96-01681-9
- Miguel A. Arcones, The law of large numbers for $U$-statistics under absolute regularity, Electron. Comm. Probab. 3 (1998), 13–19. MR 1624866, DOI 10.1214/ECP.v3-988
- Henry Berbee, Periodicity and absolute regularity, Israel J. Math. 55 (1986), no. 3, 289–304. MR 876396, DOI 10.1007/BF02765027
- Peter J. Bickel and David A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), no. 6, 1196–1217. MR 630103
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- S. Borovkova, R. Burton, and H. Dehling, Consistency of the Takens estimator for the correlation dimension, Ann. Appl. Probab. 9 (1999), no. 2, 376–390. MR 1687339, DOI 10.1214/aoap/1029962747
- Youri Davydov and Ričardas Zitikis, Convergence of generalized Lorenz curves based on stationary ergodic random sequences with deterministic noise, Statist. Probab. Lett. 59 (2002), no. 4, 329–340. MR 1935667, DOI 10.1016/S0167-7152(02)00202-X
- Youri Davydov and Ričardas Zitikis, Generalized Lorenz curves and convexifications of stochastic processes, J. Appl. Probab. 40 (2003), no. 4, 906–925. MR 2012676, DOI 10.1017/s0021900200020192
- Youri Davydov and Ričardas Zitikis, The influence of deterministic noise on empirical measures generated by stationary processes, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1203–1210. MR 2045439, DOI 10.1090/S0002-9939-03-07156-9
- Youri Davydov and Ričardas Zitikis, Convex rearrangements of random elements, Asymptotic methods in stochastics, Fields Inst. Commun., vol. 44, Amer. Math. Soc., Providence, RI, 2004, pp. 141–171. MR 2106853
- David Gilat and Roelof Helmers, On strong laws for generalized $L$-statistics with dependent data, Comment. Math. Univ. Carolin. 38 (1997), no. 1, 187–192. MR 1455483
- Gini, C. (1912). Variabilitá e mutabilita. Reprinted in: Memorie di metodologia statistica. (Ed. E. Pizetti and T. Salvemini; 1955) Libreria Eredi Virgilio Veschi, Rome.
- Charles M. Goldie, Convergence theorems for empirical Lorenz curves and their inverses, Advances in Appl. Probability 9 (1977), no. 4, 765–791. MR 478267, DOI 10.2307/1426700
- R. Helmers, Edgeworth expansions for linear combinations of order statistics, Mathematical Centre Tracts, vol. 105, Mathematisch Centrum, Amsterdam, 1982. MR 665747
- R. Helmers, P. Janssen, and R. Serfling, Glivenko-Cantelli properties of some generalized empirical DF’s and strong convergence of generalized $L$-statistics, Probab. Theory Related Fields 79 (1988), no. 1, 75–93. MR 952995, DOI 10.1007/BF00319105
- Lorenz, M. O. (1905). Methods for measuring the concentration of wealth. Amer. Stat. Assoc. 9, 209–219.
- Rosenblatt, M. (1991). Stochastic Curve Estimation. Institute of Mathematical Statistics, Hayward, CA.
- Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165, DOI 10.1002/9780470316481
- Robert J. Serfling, Generalized $L$-, $M$-, and $R$-statistics, Ann. Statist. 12 (1984), no. 1, 76–86. MR 733500, DOI 10.1214/aos/1176346393
- Shalit, H. and S. Yitzhaki, S. (1994). Marginal conditional stochastic dominance. Management Science 40, 670-684.
- Galen R. Shorack and Jon A. Wellner, Empirical processes with applications to statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 838963
- Shorrocks, A. J. (1983). Ranking income distributions. Economica 50, 3-17.
- W. R. van Zwet, A strong law for linear functions of order statistics, Ann. Probab. 8 (1980), no. 5, 986–990. MR 586781
- Shlomo Yitzhaki and Ingram Olkin, Concentration indices and concentration curves, Stochastic orders and decision under risk (Hamburg, 1989) IMS Lecture Notes Monogr. Ser., vol. 19, Inst. Math. Statist., Hayward, CA, 1991, pp. 380–392. MR 1196066, DOI 10.1214/lnms/1215459867
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
- Ričardas Zitikis
- Affiliation: Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada N6A 5B7
- Email: zitikis@stats.uwo.ca
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 3703-3712
- MSC (2000): Primary 60F15
- DOI: https://doi.org/10.1090/S0002-9939-05-08096-2
- MathSciNet review: 2163610