Refined rates of bias convergence for generalized L-Statistics in the i.i.d. case

• Autorzy: George A. Anastassiou , Tomasz Rychlik
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Bernstein operator; modulus of smoothness; generalized L-statistic; Kantorovich operator; bias; positive linear operator; random weighting; Bernstein-Durrmeyer operator; K-functional; rate of convergence

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1999
• Tom: 26
• Numer: 4
• Strony: 437-455

Daty

wydano

1999

otrzymano

1999-02-18

poprawiono

1999-06-16

Twórcy

autor

George A. Anastassiou

• Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A.

autor

Tomasz Rychlik

• Institute of Mathematics, Polish Academy of Sciences, Chopina 12 , 87-100 Toruń, Poland

Bibliografia

• G. A. Anastassiou (1993), Moments in Probability and Approximation Theory, Pitman Res. Notes Math. Ser. 287, Longman Sci. & Tech., Harlow.
• N. Balakrishnan and A. C. Cohen (1991), Order Statistics and Inference, Academic Press, Boston.
• M. Bogdan (1994), Asymptotic distributions of linear combinations of order statistics, Appl. Math. (Warsaw) 24, 201-225.
• D. Boos (1979), A differential for L-statistics, Ann. Statist. 7, 955-959.
• J. D. Cao and H. H. Gonska (1989), Pointwise estimates for modified positive linear operators, Portugal. Math. 46, 402-430.
• 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.
• H. A. David (1981), Order Statistics, 2nd ed., Wiley, New York.
• R. A. DeVore and G. G. Lorentz (1993), Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin.
• Z. Ditzian and K. Ivanov (1989), Bernstein-type operators and their derivatives, J. Approx. Theory 56, 72-90.
• 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.
• H. H. Gonska and R. K. Kovacheva (1994), The second order modulus revisited: remarks, applications, problems, Confer. Sem. Mat. Univ. Bari 257, 1-32.
• H. H. Gonska and I. Meier (1984), Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo 21, 317-335.
• 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.
• H. H. Gonska and X.-L. Zhou (1995), The strong converse inequality for the Bernstein-Kantorovich operators, ibid., 103-128.
• M. Heilmann (1988), $L_p$-saturation of some modified Bernstein operators, J. Approx. Theory 54, 260-273.
• R. Helmers, P. Janssen and R. Serfling (1990), Berry-Essen and bootstrap results for generalized L-statistics, Scand. J. Statist. 17, 65-77.
• R. Helmers and H. Ruymgaart (1988), Asymptotic normality of generalized L-statistics with unbounded scores, J. Statist. Plann. Inference 19, 43-53.
• U. Kamps (1995), A Concept of Generalized Order Statistics, Teubner Skr. Math. Stochastik, B. G. Teubner, Stuttgart.
• 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.
• C.-D. Lea and M. L. Puri (1988), Asymptotic properties of linear functions of order statistics, J. Statist. Plann. Inference 18, 203-223.
• 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.
• V. Maier (1978a), $L_p$ approximation by Kantorovich operators, Anal. Math. 4, 289-295.
• V. Maier (1978b), The $L_1$ saturation class of the Kantorovich operator, J. Approx. Theory 22, 223-232.
• D. M. Mason (1981), Asymptotic normality of linear combinations of order statistics with a smooth score function, Ann. Statist. 9, 899-908.
• D. M. Mason (1982), Some characterizations of strong laws for linear functions of order statistics, Ann. Probab. 10, 1051-1057.
• D. M. Mason and G. R. Shorack (1992), Necessary and sufficient conditions for asymptotic normality of L-statistics, ibid. 20, 1779-1804.
• R. Norvaiša (1994), Laws of large numbers for L-statistics, J. Appl. Math. Stochastic Anal. 7, 125-143.
• R. Norvaiša and R. Zitikis (1991), Asymptotic behavior of linear combinations of functions of order statistics, J. Statist. Plann. Inference 28, 305-317.
• 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.
• R. Paltanea (1998), On an optimal constant in approximation by Bernstein operators, Rend. Circ. Mat. Palermo, to appear.
• S. D. Riemenschneider (1978), The $L_p$ saturation of the Bernstein-Kantorovich polynomials, J. Approx. Theory 23, 158-162.
• 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.
• P. K. Sen (1978), An invariance principle for linear combinations of order statistics, Z. Wahrsch. Verw. Gebiete 42, 327-340.
• G. R. Shorack (1969), Asymptotic normality of linear combinations of functions of order statistics, Ann. Math. Statist. 40, 2041-2050.
• G. R. Shorack (1972), Functions of order statistics, ibid. 43, 412-427.
• L. Schumaker (1981), Spline Functions, Basic Theory, Wiley-Interscience, New York.
• P. C. Sikkema (1961), Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numer. Math. 3, 107-116.
• S. M. Stigler (1974), Linear functions of order statistics with smooth weight functions, Ann. Statist. 2, 676-693.
• S. M. Stigler (1977), Fractional order statistics, with applications, J. Amer. Statist. Assoc. 72, 544-550.
• V. Totik (1983), $L_p(p>1)$-approximation by Kantorovich polynomials, Analysis 3, 79-100.
• V. Totik (1984), An interpolation theorem and its application to positive operators, Pacific J. Math. 111, 447-481.
• J. A. Wellner (1977a), A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics, Ann. Statist. 5, 473-480.
• J. A. Wellner (1977b), A law of the iterated logarithm for functions of order statistics, ibid. 5, 481-494.
• X. Xiang (1995), A note on the bias of L-estimators and a bias reduction procedure, Statist. Probab. Lett. 23, 123-127.
• W. R. van Zwet (1980), A strong law for linear functions of order statistics, Ann. Probab. 8, 986-990.