On the strong law of large numbers for linear combinations of concomitants
DOI:
https://doi.org/10.21638/spbu01.2020.305Abstract
The article proves a theorem on the strong law of large numbers for linear functions of concomitants (induced order statistics) for sequences of independent identically distributed two-dimensional random vectors. The result complements previous work by Yang (1981), Gribkova and Zitikis (2017, 2019). The proof is based on the conditional independence property of concomitants Bhattacharya (1974), the strong law of large numbers for functions of order statistics by van Zwet (1980) is used, classical inequalities apply, including Rosenthal’s (1970).
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Литература
1. David H.A. Concomitants of order statistics // Bull. Int. Statist. Inst. 1973. Vol. 45. P. 295–300.
2. Bhattacharya P.K. Convergence of sample paths of normalized sums of induced order statistics // Ann. Statist. 1974. Vol. 2, no. 5. P. 1034–1039.
3. Bhattacharya P.K. Induced order statistics: Theory and applications. In: Handbook of Statistics / Eds. by P.R.Krishnaiah, P.K. Sen. Elsevier, 1984. Vol. 4. Nonparametric Methods. P. 383–403. https://doi.org/10.1016/S0169-7161(84)04020-7
4. David H.A., Galambos J. The asymptotic theory of concomitants of order statistics // J. Appl. Probab. 1974. Vol. 11. P. 762–770.
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6. Barnett V., Green P. J., Robinson A. Concomitants and correlation estimates // Biometrika. 1976. Vol. 63. P. 323–328.
7. Wang X., Stokes S. L., Lim J., Chen M. Concomitants of multivariate order statistics with application to judgment post-stratification // J. Amer. Statist. Assoc. 2006. Vol. 101. P. 1693–1704.
8. Ke Wang M. S. On concomitants of order statistics. PhD thesis. The Ohio State University, 2008.
9. Егоров В.А., Невзоров В.Б. Некоторые теоремы для индуцированных порядковых статистик // Теория вероятн. и ее примен. 1982. Т. 27, №3. С. 592–599.
10. Егоров В.А., Невзоров В.Б. О скорости сходимости к нормальному закону сумм индуцированных порядковых статистик // Зап. научн. сем. ЛОМИ. 1981. Т. 108. P. 45–56.
11. Davydov Y., Egorov V. Functional limit theorems for induced order statistics // Math. Meth. Stat. 2000. Vol. 9. P. 297–313.
12. Davydov Y., Egorov V. Functional CLT and LIL for induced order statistics. In: Asymptotic Methods in Probability and Statistics with Applications / Eds. by N.Balakrishnan, I.A. Ibragimov, V.B.Nevzorov. Boston, MA: Birkh¨auser, 2001. P. 333–349 (Statistics for Industry and Technology). https://doi.org/10.1007/978-1-4612-0209-7_24
13. Yang S.-S. General distribution theory of the concomitants of order statistics // Ann. Statist. 1977. Vol. 5. P. 996–1002.
14. Yang S.-S. Linear combination of concomitants of order statistics with application to testing and estimation // Ann. Inst. Statist. Math. 1981. Vol. 33, no. 3. P. 463–470.
15. Yang S.-S. Linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function // J. Amer. Statist. Assoc. 1981. Vol. 76. P. 658–662.
16. Gribkova N., Zitikis R. Statistical foundations for assessing the difference between the classical and weighted-Gini betas // Math. Meth. Stat. 2017. Vol. 26, no. 4. P. 267–281. https://doi.org/10.3103/S1066530717040020
17. Gribkova N., Zitikis R. Weighted allocations, their concomitant-based estimators, and asymptotics // Ann. Inst. Statist. Math. 2019. Vol. 71, no. 4. P. 811–835. https://doi.org/10.1007/s10463-018-0660-2
18. Rosenthal H. P. On the subspaces of Lp (p > 2) spanned by sequences of independent random variables // Israel J. Math. 1970. Vol. 8, no. 3. P. 273–303.
19. van Zwet W.R. A strong law for linear functions of order statistics // Ann. Probab. 1980. Vol. 8, no. 5. P. 986–990.
References
1. David H.A., “Concomitants of order statistics”, Bull. Int. Statist. Inst. 45, 295–300 (1973).
2. Bhattacharya P.K., “Convergence of sample paths of normalized sums of induced order statistics”, Ann. Statist. 2 (5), 1034–1039 (1974).
3. Bhattacharya P.K., “Induced order statistics: Theory and applications”, in: Handbook of Statistics 4, 383–403 (P.R.Krishnaiah, P.K. Sen (eds.), Elsevier, 1984). https://doi.org/10.1016/S0169-7161(84)04020-7
4. David H.A., Galambos J., “The asymptotic theory of concomitants of order statistics”, J. Appl. Probab. 11, 762–770 (1974).
5. David H.A., Nagaraja H.N., “Concomitants of order statistics”, in: Handbook of Statistics 16, 487–513 (N.Balakrishnan, C.R.Rao (eds.), Elsevier, 1998). https://doi.org/10.1016/S0169-7161(98)16020-0
6. Barnett V., Green P. J., Robinson A., “Concomitants and correlation estimates”, Biometrika 63, 323–328 (1976).
7. Wang X., Stokes S. L., Lim J., Chen M., “Concomitants of multivariate order statistics with application to judgment post-stratification”, J. Amer. Statist. Assoc. 101, 1693–1704 (2006).
8. Ke Wang M. S., On concomitants of order statistics (PhD thesis, The Ohio State University, 2008).
9. Egorov V.A., Nevzorov V.B., “Some theorems for induced order statistics”, Theory Probab. Appl. 27 (3), 633–639 (1983). https://doi.org/10.1137/1127074
10. Egorov V.A., Nevzorov V.B., “Rate of convergence to the normal law of sums of induced order statistics”, J. Math. Sci. 25 (3), 1139–1146 (1984). https://doi.org/10.1007/BF01084792
11. Davydov Y., Egorov V., “Functional limit theorems for induced order statistics”, Math. Meth. Stat. 9, 297–313 (2000).
12. Davydov Y., Egorov V., “Functional CLT and LIL for induced order statistics”, in: Asymptotic Methods in Probability and Statistics with Applications, 333–349 (N.Balakrishnan, I.A. Ibragimov, V.B.Nevzorov (eds.), Birkh¨auser, Boston, MA, 2001, Statistics for Industry and Technology). https://doi.org/10.1007/978-1-4612-0209-7_24
13. Yang S.-S., “General distribution theory of the concomitants of order statistics”, Ann. Statist. 5, 996–1002 (1977).
14. Yang S.-S., “Linear combination of concomitants of order statistics with application to testing and estimation”, Ann. Inst. Statist. Math. 33 (3), 463–470 (1981).
15. Yang S.-S., “Linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function”, J. Amer. Statist. Assoc. 76, 658–662 (1981).
16. Gribkova N., Zitikis R., “Statistical foundations for assessing the difference between the classical and weighted-Gini betas”, Math. Meth. Stat. 26 (4), 267–281 (2017). https://doi.org/10.3103/S1066530717040020
17. Gribkova N., Zitikis R., “Weighted allocations, their concomitant-based estimators, and asymptotics”, Ann. Inst. Statist. Math. 71 (4), 811–835 (2019). https://doi.org/10.1007/s10463-018-0660-2
18. Rosenthal H.P., “On the subspaces of Lp (p > 2) spanned by sequences of independent random variables”, Israel J. Math. 8 (3), 273–303 (1970).
19. van Zwet W.R., “A strong law for linear functions of order statistics”, Ann. Probab. 8 (5), 986–990 (1980).
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Published
2020-09-04
How to Cite
Dudkina, O. I., & Gribkova, N. V. (2020). On the strong law of large numbers for linear combinations of concomitants. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(3), 418–424. https://doi.org/10.21638/spbu01.2020.305
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
