Об усиленном законе больших чисел для линейных комбинаций конкомитантов

Авторы

  • Ольга Игоревна Дудкина
  • Надежда Викторовна Грибкова

DOI:

https://doi.org/10.21638/spbu01.2020.305

Аннотация

В статье доказана теорема об усиленном законе больших чисел для линейных функций конкомитантов (индуцированных порядковых статистик) для последовательностей независимых одинаково распределенных двумерных случайных векторов. Результат дополняет предшествующие работы Янга (1981), Грибковой и Зитикиса (2017, 2019). Доказательство базируется на свойстве условной независимости конкомитантов Бхаттачариа (1974), применяется усиленный закон больших чисел для функций порядковых статистик ван Цвета (1980), классические неравенства, в том числе неравенство Розенталя (1970).

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Библиографические ссылки

Литература

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6. Barnett V., Green P. J., Robinson A. Concomitants and correlation estimates // Biometrika. 1976. Vol. 63. P. 323–328.

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8. Ke Wang M. S. On concomitants of order statistics. PhD thesis. The Ohio State University, 2008.

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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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Опубликован

04.09.2020

Как цитировать

Дудкина, О. И., & Грибкова, Н. В. (2020). Об усиленном законе больших чисел для линейных комбинаций конкомитантов. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия, 7(3), 418–424. https://doi.org/10.21638/spbu01.2020.305

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Математика