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.
@article{bwmeta1.element.bwnjournal-article-zmv22z2p201bwm,
author = {Ma\l gorzata Bogdan},
title = {Asymptotic distributions $\omicron$f linear combinations of order statistics},
journal = {Applicationes Mathematicae},
volume = {22},
year = {1994},
pages = {201-225},
zbl = {0806.62012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv22z2p201bwm}
}
Bogdan, Małgorzata. Asymptotic distributions οf linear combinations of order statistics. Applicationes Mathematicae, Tome 22 (1994) pp. 201-225. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv22z2p201bwm/
[000] [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. | Zbl 0614.62015
[001] [2] S. Bjerve, Error bounds for linear combinations of order statistics, ibid. 5 (1977), 357-369. | Zbl 0356.62015
[002] [3] N. Bleinstein, Uniform expansions of integrals with stationary points near algebraic singularity, Comm. Pure Appl. Math. 19 (1966), 353-370. | Zbl 0145.15801
[003] [4] D. D. Boos, The differential approach in statistical theory and robust inference, PhD thesis, Florida State University, 1977.
[004] [5] D. D. Boos, A differential for L-statistics, Ann. Statist. 7 (1979), 955-959.
[005] [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.
[006] [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. | Zbl 0157.47701
[007] [8] H. E. Daniels, Saddlepoint approximations in statistics, ibid. 25 (1954), 631-649. | Zbl 0058.35404
[008] [9] H. E. Daniels, Tail probability approximations, Internat. Statist. Rev. 55 (1987), 37-48. | Zbl 0614.62016
[009] [10] G. Easton and E. Ronchetti, General saddlepoint approximations, J. Amer. Statist. Assoc. 81 (1986), 420-430. | Zbl 0611.62014
[010] [11] R. Helmers, Edgeworth expansions for linear combinations of order statistics with smooth weight functions, Ann. Statist. 8 (1980), 1361-1374. | Zbl 0444.62053
[011] [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. | Zbl 0786.60027
[012] [13] J. L. Jensen, Uniform saddlepoint approximations, Adv. Appl. Probab. 20 (1988), 622-634. | Zbl 0656.60043
[013] [14] R. Lugannani and S. Rice, Saddlepoint approximations for the distribution of the sum of independent random variables, ibid. 12 (1980), 475-490. | Zbl 0425.60042
[014] [15] N. Reid, Saddlepoint methods and statistical inference, Statist. Sci. 3 (1988), 213-238. | Zbl 0955.62541
[015] [16] R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley, 1980. | Zbl 0538.62002
[016] [17] G. R. Shorack, Asymptotic normality of linear combinations of order statistic, Ann. Math. Statist. 40 (1969), 2041-2050. | Zbl 0188.51002
[017] [18] G. R. Shorack, Functions of order statistics, ibid. 43 (1972), 412-427. | Zbl 0239.62037
[018] [19] S. M. Stigler, Linear functions of order statistics, ibid. 40 (1969), 770-788. | Zbl 0186.52502
[019] [20] S. M. Stigler, The asymptotic distribution of the trimmed mean, Ann. Statist. 1 (1973), 472-477. | Zbl 0261.62016
[020] [21] S. M. Stigler, Linear functions of order statistics with smooth weight functions, ibid. 2 (1974), 676-693. | Zbl 0286.62028
[021] [22] R. von Mises, On the asymptotic distribution of differentiable statistical functions, Ann. Math. Statist. 18 (1947), 309-348. | Zbl 0037.08401