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On functionals of order statistics

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Let gn be real functions,U ni, 1≤in, the ordered sample ofn independentU(0,1) distributed random variables, andc ni(α), 1≤i≤n, 0≤α≤1 be (known) real numbers,n=1, 2, ... The random quantity\(T_n (\alpha ): = n^{ - 1} \sum\limits_{i = 1}^n {c_{ni} (\alpha )g_n (U_{ni} )} \), 0≤α≤1, is studied. Based on a method proposed byShorack [1972] the main result is the weak convergence of\(H_n : = n^{1/2} (T_n - \mu _n )\) to Gaussian processes, where\(\mu _n (\alpha ): = \sum\limits_{i = 1}^n {c_{ni} (\alpha )} \int\limits_{(i - 1)/n}^{i/n} {g_n (t)dt} \), 0≤α≤1. The convergence is with respect to theSkorokhod [1956]-topologiesM 2,M 1 onD (I) and the ‖ ‖-topology onC(I), depending on the conditions imposed on thec ni(α).

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  1. Bickel, P.J.: Some contributions to the theory of order statistics. Proc. Fifth Berkeley Symp. Math. Stat. Prob., Vol. 1, 1967.

  2. Bickel, P.J., andM.J. Wichura: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat.42, 1971, 1656–1670.

  3. Billingsley, P.: Convergence of probability measures. New York 1968.

  4. Chandra, M., andN.D. Singpurwalla: The Gini-index, the Lorenz curve, and the total time on test transforms. Inst. f. Management Sci., Washington, preprint, 1978.

  5. Chernoff, H., J.L. Gastwirth, andM.V. Johns Jr.: Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Ann. Math. Stat.33, 1967, 52–72.

  6. Gänssler, P., andW. Stute: Wahrscheinlichkeitstheorie. Berlin 1977.

  7. Gikhman, I.I., andA.V. Skorokhod: Introduction to the theory of random processes. Philadelphia 1969.

  8. Gikhman, I.I., andA.V. Skorokhod: The theory of stochastic processes. Berlin 1974.

  9. Goldie, C.: Convergence theorems for empirical Lorenz-curves and their inverses. Adv. Appl. Prob.9, 1977, 765–791.

  10. Govindarajulu, Z.: Asymptotic normality of linear combinations of order statistics. II. Proc. Nat. Acad. Sci.59, 1968, 713–719.

  11. Hecker, H.: A characterization of the asymptotic normality of linear combinations of order statistics from the uniform distribution. Ann. Stat.4, 1975, 1244–1246.

  12. Hoffman-Jørgenson, J.: The Theory of Analytic Spaces. Aarhus, Series No. 10, 1970.

  13. Moore, D.S.: An elementary proof of asymptotic normality of linear functions of order statistics. Ann. Math. Stat.39, 1968, 263–265.

  14. Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat.42, 1971, 1285–1295.

  15. Parthasarathy, K.R.: Probability measures on metric spaces. New York-London 1967.

  16. Pyke, R., andG.R. Shorack: Weak convergence of two-sample empirical process and a new approach to Chernoff-Savage theorems. Ann. Math. Stat.39, 1968, 755–771.

  17. Sendler, W.: On statistical inference in concentration measurement. Metrika26, 1979, 109–122.

  18. Shorack, G.R.: Functions of order statistics. Ann. Math. Stat.43, 1972, 412–427.

  19. Skorokhod, A.V.: Limit theorems for stochastic processes. Theory Prob. Appl.1, 1956, 261–290.

  20. Stigler, S.M.: The asymptotic distribution of the trimmed mean. Ann. Stat.1, 1973, 472–477.

  21. Wichura, M.: On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Stat.41, 1970, 284–291.

  22. Zwet, W.v.: A strong law for linear functions of order statistics. Ann. Prob.8, 1980, 986–990.

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Sendler, W. On functionals of order statistics. Metrika 29, 19–54 (1982). https://doi.org/10.1007/BF01893363

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  • Real Number
  • Stochastic Process
  • Probability Theory
  • Economic Theory
  • Order Statistic