Asymptotic distributions οf linear combinations of order statistics

Małgorzata Bogdan

Applicationes Mathematicae (1994)

  • Volume: 22, Issue: 2, page 201-225
  • ISSN: 1233-7234

Abstract

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We study the asymptotic distributions of linear combinations of order statistics (L-statistics) which can be expressed as differentiable statistical functionals and we obtain Berry-Esseen type bounds and the Edgeworth series for the distribution functions of L-statistics. We also analyze certain saddlepoint approximations for the distribution functions of L-statistics.

How to cite

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Bogdan, Małgorzata. "Asymptotic distributions οf linear combinations of order statistics." Applicationes Mathematicae 22.2 (1994): 201-225. <http://eudml.org/doc/219092>.

@article{Bogdan1994,
abstract = {We study the asymptotic distributions of linear combinations of order statistics (L-statistics) which can be expressed as differentiable statistical functionals and we obtain Berry-Esseen type bounds and the Edgeworth series for the distribution functions of L-statistics. We also analyze certain saddlepoint approximations for the distribution functions of L-statistics.},
author = {Bogdan, Małgorzata},
journal = {Applicationes Mathematicae},
keywords = {Berry-Esseen type bounds; saddlepoint method; Edgeworth series; von Mises representation; statistical function; L-statistic; L-statistics; order statistics; Edgeworth expansion; saddlepoint approximations},
language = {eng},
number = {2},
pages = {201-225},
title = {Asymptotic distributions οf linear combinations of order statistics},
url = {http://eudml.org/doc/219092},
volume = {22},
year = {1994},
}

TY - JOUR
AU - Bogdan, Małgorzata
TI - Asymptotic distributions οf linear combinations of order statistics
JO - Applicationes Mathematicae
PY - 1994
VL - 22
IS - 2
SP - 201
EP - 225
AB - We study the asymptotic distributions of linear combinations of order statistics (L-statistics) which can be expressed as differentiable statistical functionals and we obtain Berry-Esseen type bounds and the Edgeworth series for the distribution functions of L-statistics. We also analyze certain saddlepoint approximations for the distribution functions of L-statistics.
LA - eng
KW - Berry-Esseen type bounds; saddlepoint method; Edgeworth series; von Mises representation; statistical function; L-statistic; L-statistics; order statistics; Edgeworth expansion; saddlepoint approximations
UR - http://eudml.org/doc/219092
ER -

References

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  1. [1] P. J. Bickel, F. Götze, and W. R. van Zwet, The Edgeworth expansion for U-statistics of degree two, Ann. Statist. 14 (1986), 1463-1484. Zbl0614.62015
  2. [2] S. Bjerve, Error bounds for linear combinations of order statistics, ibid. 5 (1977), 357-369. Zbl0356.62015
  3. [3] N. Bleinstein, Uniform expansions of integrals with stationary points near algebraic singularity, Comm. Pure Appl. Math. 19 (1966), 353-370. Zbl0145.15801
  4. [4] D. D. Boos, The differential approach in statistical theory and robust inference, PhD thesis, Florida State University, 1977. 
  5. [5] D. D. Boos, A differential for L-statistics, Ann. Statist. 7 (1979), 955-959. 
  6. [6] D. D. Boos and R. J. Serfling, On Berry-Esseen rates for statistical functions, with applications to L-estimates, technical report, Florida State Univ., 1979. 
  7. [7] H. Chernoff, J. L. Gastwirth, and V. M. Johns Jr., Asymptotic distribution of linear combinations of order statistics with application to estimations, Ann. Math. Statist. 38 (1967), 52-72. Zbl0157.47701
  8. [8] H. E. Daniels, Saddlepoint approximations in statistics, ibid. 25 (1954), 631-649. Zbl0058.35404
  9. [9] H. E. Daniels, Tail probability approximations, Internat. Statist. Rev. 55 (1987), 37-48. Zbl0614.62016
  10. [10] G. Easton and E. Ronchetti, General saddlepoint approximations, J. Amer. Statist. Assoc. 81 (1986), 420-430. Zbl0611.62014
  11. [11] R. Helmers, Edgeworth expansions for linear combinations of order statistics with smooth weight functions, Ann. Statist. 8 (1980), 1361-1374. Zbl0444.62053
  12. [12] T. Inglot and T. Ledwina, Moderately large deviations and expansions of large deviations for some functionals of weighted empirical process, Ann. Probab., to appear. Zbl0786.60027
  13. [13] J. L. Jensen, Uniform saddlepoint approximations, Adv. Appl. Probab. 20 (1988), 622-634. Zbl0656.60043
  14. [14] R. Lugannani and S. Rice, Saddlepoint approximations for the distribution of the sum of independent random variables, ibid. 12 (1980), 475-490. Zbl0425.60042
  15. [15] N. Reid, Saddlepoint methods and statistical inference, Statist. Sci. 3 (1988), 213-238. Zbl0955.62541
  16. [16] R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley, 1980. Zbl0538.62002
  17. [17] G. R. Shorack, Asymptotic normality of linear combinations of order statistic, Ann. Math. Statist. 40 (1969), 2041-2050. Zbl0188.51002
  18. [18] G. R. Shorack, Functions of order statistics, ibid. 43 (1972), 412-427. Zbl0239.62037
  19. [19] S. M. Stigler, Linear functions of order statistics, ibid. 40 (1969), 770-788. Zbl0186.52502
  20. [20] S. M. Stigler, The asymptotic distribution of the trimmed mean, Ann. Statist. 1 (1973), 472-477. Zbl0261.62016
  21. [21] S. M. Stigler, Linear functions of order statistics with smooth weight functions, ibid. 2 (1974), 676-693. Zbl0286.62028
  22. [22] R. von Mises, On the asymptotic distribution of differentiable statistical functions, Ann. Math. Statist. 18 (1947), 309-348. Zbl0037.08401

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