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The exact and approximate distributions of linear combinations of selected order statistics from a uniform distribution

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Summary

The exact probability density function is given for linear combinations ofk=k(n) order statistics selected from whole order statistics based on random sample of sizen drawn from a uniform distribution. Normal approximation to the linear combinations is made with the aid of Berry-Esseen's theorem. Necessary and sufficient conditions of the asymptotic normality for the statistic are obtained, too. An exact distribution and its normal approximation of linear combination of mutually independent gamma variables with integer valued parameters are also given as associated consequences.

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References

  1. Andrews, G. E. (1976).The Theory of Partitions, Encyclopedia of Mathematics and Its Applications 2, Addison-Wesley.

  2. Chernoff, H., Gastwirth, J. L. and Johns, M. V. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimations,Ann. Math. Statist.,38, 52–72.

    Article  MathSciNet  Google Scholar 

  3. Dempster, A. P. and Kleyle, R. M. (1968). Distributions determined by cutting a simplex with hyperplanes,Ann. Math. Statist.,39, 1473–1478.

    Article  MathSciNet  Google Scholar 

  4. Doetsch, G. (1961).Guide to the Applications of Laplace Transforms, D. Van Nostrand.

  5. Eicker, F. and Puri, M. L. (1976). On linear combination of order statistics,Essays in Probability and Statistics, Shinko Tsusho Co., Ltd. (ed. S. Ikeda and others), 433–449.

  6. Hecker, H. (1976). A characterization of the asymptotic normality of linear combinations of order statistics from the uniform distribution,Ann. Statist.,4, 1244–1246.

    Article  MathSciNet  Google Scholar 

  7. Matsunawa, T. (1982). Some strong ε-equivalence of random variables,Ann. Inst. Statist. Math.,34, A, 209–224.

    Article  MathSciNet  Google Scholar 

  8. Pyke, R. (1965). Spacings,J. Roy. Statist. Soc., B,27, 395–436.

    MathSciNet  MATH  Google Scholar 

  9. Shorack, G. (1969). Asymptotic normality of linear combinations of functions of order statistics,Ann. Math. Statist.,40, 2041–2050.

    Article  MathSciNet  Google Scholar 

  10. Stigler, S. M. (1974). Linear functions of order statistics with smooth weight functions,Ann. Statist.,4, 676–693.

    Article  MathSciNet  Google Scholar 

  11. van Beek, P. (1972). An application of Fourier Methods to the problem of sharpning the Berry-Esseen inequality,Zeit. Wahrsheinlichkeitsth.,23, 187–196.

    Article  Google Scholar 

  12. van Zwet, W. R. (1979). The Edgeworth expansion for linear combinations of uniform order statistics,Proceedings of the Second Prague Symposium on Asymptotic Statistics (ed. Mandl, P. and Hušková), 93–101, North-Holland.

  13. Weisberg, H. (1971). The distribution of linear combinations of order statistics from the uniform distribution,Ann. Math. Statist.,42, 704–709.

    Article  MathSciNet  Google Scholar 

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Authors
  1. T. Matsunawa
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The Institute of Statistical Mathematics

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Matsunawa, T. The exact and approximate distributions of linear combinations of selected order statistics from a uniform distribution. Ann Inst Stat Math 37, 1–16 (1985). https://doi.org/10.1007/BF02481076

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  • DOI: https://doi.org/10.1007/BF02481076

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