Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On functionals of order statistics

  • 29 Accesses

  • 6 Citations

Summary

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).

This is a preview of subscription content, log in to check access.

References

  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.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

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