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Evaluations of generalized order statistics from bounded populations

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Abstract

We consider generalized order statistics with arbitrary model parameters based on distributions supported on finite intervals. We determine optimal bounds on the expectations of arbitrary linear combinations of centered generalized order statistics gauged in support length scale units. More precise representations of bounds are obtained for single generalized order statistics and respective differences.

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References

  • Aggarwala R, Balakrishnan N (2000) Progressive censoring. Theory, methods and applications. Birkhäuser, Boston

    Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New York

    MATH  Google Scholar 

  • Balakrishnan N, Cramer E, Kamps U (2001) Bounds for means and variances of progressively type II censored order statistics. Stat Probab Lett 54: 301–315

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan N, Dembińska A (2008) Progressively type-II censored order statistics from discrete distributions. J Stat Plann Inf 138: 845–856

    Article  MATH  Google Scholar 

  • Bieniek M (2006) Projection bounds on expectations of generalized order statistics from DFR and DFRA families. Statistics 40: 339–351

    MATH  MathSciNet  Google Scholar 

  • Bieniek M (2007a) Variation diminishing property of densities of uniform generalized order statistics. Metrika 65: 297–309

    Article  MathSciNet  Google Scholar 

  • Bieniek M (2007b) Bounds for expectations of differences of generalized order statistics based on general and life distributions. Commun Stat Theory Methods 36: 59–72

    Article  MATH  MathSciNet  Google Scholar 

  • Bieniek M (2007c) Projection bounds on expectations of spacings of generalized order statistics from DD and DDA families. Commun Stat Theory Methods 36: 1343–1357

    Article  MATH  MathSciNet  Google Scholar 

  • Bieniek M (2008a) Projection bounds on expectations of generalized order statistics from DD and DDA families. J Stat Plann Inf 138: 971–981

    Article  MATH  MathSciNet  Google Scholar 

  • Bieniek M (2008b) Projection bounds on expectations of spacings of generalized order statistics from DFR and DFRA families. Statistics 42: 231–243

    Article  MATH  MathSciNet  Google Scholar 

  • Bieniek M (2008c) On families of distributions for which optimal bounds on expectations of GOS can be derived. Commun Stat Theory Methods 37: 1997–2009

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E, Kamps U (2001) On distributions of generalized order statistics. Statistics 35: 269–280

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E, Kamps U, Rychlik T (2002) Evaluations of expected generalized order statistics in various scale units. Appl Math (Warsaw) 29: 285–295

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E, Kamps U, Rychlik T (2004) Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann Inst Stat Math 56: 183–192

    Article  MATH  MathSciNet  Google Scholar 

  • David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, Hoboken

    MATH  Google Scholar 

  • Kamps U (1995) A concept of generalized order statistics. Teubner, Stuttgart

    MATH  Google Scholar 

  • Kamps U (1999) Order statistics, generalized. In: Kotz S, Balakrishnan N, Read C, Vidakovic B (eds) Encyclopedia of statistical sciences, update vol 3. Wiley, New York, pp 553–557

    Google Scholar 

  • Klimczak M (2007) Best bounds for kth records from bounded samples. Commun Stat Theory Methods 36: 1451–1464

    Article  MATH  MathSciNet  Google Scholar 

  • Mathai AM (1993) A handbook of generalized special functions for statistical and physical sciences. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Moriguti S (1953) A modification of Schwarz’s inequality, with applications to distributions. Ann Math Stat 24: 107–113

    Article  MATH  MathSciNet  Google Scholar 

  • Raqab MZ (2003) P-norm bounds for moments of progressively type II censored order statistics. Stat Probab Lett 64: 393–402

    Article  MATH  MathSciNet  Google Scholar 

  • Rychlik T (2001) Projecting statistical functionals. Lecture Notes in Statistics, vol 160. Springer, New York

    Google Scholar 

  • Rychlik T (2006) Best bounds on expectations of L-statistics from bounded samples. In: Balakrishnan N, Castillo E, Sarabia J-M (eds) Advances in distribution theory, order statistics and inference. Birkhäuser, Boston, pp 253–263

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Chopina 12, 87100, Toruń, Poland

    Tomasz Rychlik

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  1. Tomasz Rychlik
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Correspondence to Tomasz Rychlik.

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Rychlik, T. Evaluations of generalized order statistics from bounded populations. Stat Papers 51, 165–177 (2010). https://doi.org/10.1007/s00362-008-0129-0

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  • DOI: https://doi.org/10.1007/s00362-008-0129-0

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