Using tools of approximation theory, we evaluate rates of bias convergence for sequences of generalized L-statistics based on i.i.d. samples under mild smoothness conditions on the weight function and simple moment conditions on the score function. Apart from standard methods of weighting, we introduce and analyze L-statistics with possibly random coefficients defined by means of positive linear functionals acting on the weight function.
@article{bwmeta1.element.bwnjournal-article-zmv26i4p437bwm,
author = {George Anastassiou and Tomasz Rychlik},
title = {Refined rates of bias convergence for generalized L-Statistics in the i.i.d. case},
journal = {Applicationes Mathematicae},
volume = {26},
year = {1999},
pages = {437-455},
zbl = {0993.62042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i4p437bwm}
}
Anastassiou, George; Rychlik, Tomasz. Refined rates of bias convergence for generalized L-Statistics in the i.i.d. case. Applicationes Mathematicae, Tome 26 (1999) pp. 437-455. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i4p437bwm/
[000] G. A. Anastassiou (1993), Moments in Probability and Approximation Theory, Pitman Res. Notes Math. Ser. 287, Longman Sci. & Tech., Harlow. | Zbl 0847.41001
[001] N. Balakrishnan and A. C. Cohen (1991), Order Statistics and Inference, Academic Press, Boston. | Zbl 0732.62044
[002] M. Bogdan (1994), Asymptotic distributions of linear combinations of order statistics, Appl. Math. (Warsaw) 24, 201-225. | Zbl 0806.62012
[003] D. Boos (1979), A differential for L-statistics, Ann. Statist. 7, 955-959. | Zbl 0423.62021
[004] J. D. Cao and H. H. Gonska (1989), Pointwise estimates for modified positive linear operators, Portugal. Math. 46, 402-430. | Zbl 0683.41031
[005] H. Chernoff, J. L. Gastwirth and M. V. Johns (1967), Asymptotic distribution of linear combinations of order statistics, with applications to estimation, Ann. Math. Statist. 38, 52-72. | Zbl 0157.47701
[006] H. A. David (1981), Order Statistics, 2nd ed., Wiley, New York. | Zbl 0553.62046
[007] R. A. DeVore and G. G. Lorentz (1993), Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin.
[008] Z. Ditzian and K. Ivanov (1989), Bernstein-type operators and their derivatives, J. Approx. Theory 56, 72-90. | Zbl 0692.41021
[009] I. Gavrea and D. H. Mache (1995), Generalization of Bernstein-type approximation methods, in: Approximation Theory, Proc. IDoMAT95, M. W. Müller, M. Felten and D. H. Mache (eds.), Math. Res. 86, Akademie-Verlag, Berlin, 115-126. | Zbl 1005.41500
[010] H. H. Gonska and R. K. Kovacheva (1994), The second order modulus revisited: remarks, applications, problems, Confer. Sem. Mat. Univ. Bari 257, 1-32. | Zbl 1009.41012
[011] H. H. Gonska and I. Meier (1984), Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo 21, 317-335. | Zbl 0568.41021
[012] H. H. Gonska and D.-X. Zhou (1995), Local smoothness of functions and Bernstein-Durrmeyer operators, Comput. Math. Appl. 30, No. 3-6 (special issue Concrete Analysis, G. A. Anastassiou (ed.)), 83-101. | Zbl 0838.41016
[013] H. H. Gonska and X.-L. Zhou (1995), The strong converse inequality for the Bernstein-Kantorovich operators, ibid., 103-128. | Zbl 0842.41019
[014] M. Heilmann (1988), -saturation of some modified Bernstein operators, J. Approx. Theory 54, 260-273. | Zbl 0654.41021
[015] R. Helmers, P. Janssen and R. Serfling (1990), Berry-Essen and bootstrap results for generalized L-statistics, Scand. J. Statist. 17, 65-77. | Zbl 0706.62016
[016] R. Helmers and H. Ruymgaart (1988), Asymptotic normality of generalized L-statistics with unbounded scores, J. Statist. Plann. Inference 19, 43-53. | Zbl 0664.62048
[017] U. Kamps (1995), A Concept of Generalized Order Statistics, Teubner Skr. Math. Stochastik, B. G. Teubner, Stuttgart.
[018] H.-B. Knoop and X.-L. Zhou (1992), The lower estimate for linear positive operators, part 1: Constr. Approx. 11 (1995), 53-66, part 2: Results Math. 25 (1994), 300-315.
[019] C.-D. Lea and M. L. Puri (1988), Asymptotic properties of linear functions of order statistics, J. Statist. Plann. Inference 18, 203-223. | Zbl 0652.62015
[020] D. H. Mache (1995), A link between Bernstein polynomials and Durrmeyer polynomials with Jacobi weights, in: Approximation Theory VIII, Vol. 1: Approximation and Interpolation, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 403-410. | Zbl 1137.41336
[021] V. Maier (1978a), approximation by Kantorovich operators, Anal. Math. 4, 289-295. | Zbl 0435.41011
[022] V. Maier (1978b), The saturation class of the Kantorovich operator, J. Approx. Theory 22, 223-232. | Zbl 0397.41012
[023] D. M. Mason (1981), Asymptotic normality of linear combinations of order statistics with a smooth score function, Ann. Statist. 9, 899-908. | Zbl 0472.62057
[024] D. M. Mason (1982), Some characterizations of strong laws for linear functions of order statistics, Ann. Probab. 10, 1051-1057. | Zbl 0505.60032
[025] D. M. Mason and G. R. Shorack (1992), Necessary and sufficient conditions for asymptotic normality of L-statistics, ibid. 20, 1779-1804. | Zbl 0765.62024
[026] R. Norvaiša (1994), Laws of large numbers for L-statistics, J. Appl. Math. Stochastic Anal. 7, 125-143. | Zbl 0823.60028
[027] R. Norvaiša and R. Zitikis (1991), Asymptotic behavior of linear combinations of functions of order statistics, J. Statist. Plann. Inference 28, 305-317. | Zbl 0732.62046
[028] R. Paltanea (1995), Best constants in estimates with second order moduli of continuity, in: Approximation Theory, Proc. IDoMAT95, M. W. Müller, M. Felten and D. H. Mache (eds.), Math. Res. 86, Akademie-Verlag, Berlin, 251-275. | Zbl 0839.41019
[029] R. Paltanea (1998), On an optimal constant in approximation by Bernstein operators, Rend. Circ. Mat. Palermo, to appear. | Zbl 0905.41009
[030] S. D. Riemenschneider (1978), The saturation of the Bernstein-Kantorovich polynomials, J. Approx. Theory 23, 158-162. | Zbl 0388.41011
[031] V. K. Rohatgi and A. K. M. D. E. Saleh (1988), A class of distributions connected to order statistics with nonintegral sample size, Comm. Statist. Theory Methods 17, 2005-2012. | Zbl 0639.62008
[032] P. K. Sen (1978), An invariance principle for linear combinations of order statistics, Z. Wahrsch. Verw. Gebiete 42, 327-340. | Zbl 0362.60022
[033] G. R. Shorack (1969), Asymptotic normality of linear combinations of functions of order statistics, Ann. Math. Statist. 40, 2041-2050. | Zbl 0188.51002
[034] G. R. Shorack (1972), Functions of order statistics, ibid. 43, 412-427. | Zbl 0239.62037
[035] L. Schumaker (1981), Spline Functions, Basic Theory, Wiley-Interscience, New York. | Zbl 0449.41004
[036] P. C. Sikkema (1961), Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numer. Math. 3, 107-116. | Zbl 0099.04701
[037] S. M. Stigler (1974), Linear functions of order statistics with smooth weight functions, Ann. Statist. 2, 676-693. | Zbl 0286.62028
[038] S. M. Stigler (1977), Fractional order statistics, with applications, J. Amer. Statist. Assoc. 72, 544-550. | Zbl 0374.62048
[039] V. Totik (1983), -approximation by Kantorovich polynomials, Analysis 3, 79-100. | Zbl 0501.41025
[040] V. Totik (1984), An interpolation theorem and its application to positive operators, Pacific J. Math. 111, 447-481. | Zbl 0501.41003
[041] J. A. Wellner (1977a), A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics, Ann. Statist. 5, 473-480. | Zbl 0365.62045
[042] J. A. Wellner (1977b), A law of the iterated logarithm for functions of order statistics, ibid. 5, 481-494. | Zbl 0365.62046
[043] X. Xiang (1995), A note on the bias of L-estimators and a bias reduction procedure, Statist. Probab. Lett. 23, 123-127. | Zbl 0819.62030
[044] W. R. van Zwet (1980), A strong law for linear functions of order statistics, Ann. Probab. 8, 986-990. | Zbl 0448.60025